CELE Engineering Mathematics — Algebra and FundamentalsMemory Anchors

Tags

• formula
• quadratic
• roots
• MSTE

Topic

Quadratic Equations

Concept

Quadratic Formula: x = [-b ± √(b²−4ac)] / 2a

Anchor Id

A1

Difficulty

easy

Memory Aid

Sing it to the tune of 'Pop Goes the Weasel': 'x equals negative b, plus or minus the square root, of b-squared minus four-a-c, all over two-a!' Philippine board reviewees have been singing this for decades — it works because the melody acts as a built-in retrieval cue. Every time you hear a pop song, you remember the quadratic formula.

Anchor Type

rhyme

Why It Works

Musical mnemonics exploit the brain's procedural memory system. The rhythm and melody encode the formula sequence in a different neural pathway than rote memorization, making it more durable under stress.

Example Usage

Board problem: Solve 2x² + 3x − 5 = 0. Sing the tune: a=2, b=3, c=−5. x = [−3 ± √(9+40)] / 4 = [−3 ± 7] / 4. Roots: x = 1 or x = −5/2.

Recall Trigger

Hear the tune of 'Pop Goes the Weasel' or the phrase 'negative b'

Tags

• discriminant
• roots
• nature of roots
• quadratic

Topic

Quadratic Equations

Concept

Discriminant determines the nature of roots: b²−4ac > 0 (real distinct), = 0 (repeated), < 0 (complex)

Anchor Id

A2

Difficulty

easy

Memory Aid

Think of the discriminant as a JEEPNEY FARE CHECK. Positive discriminant = two passengers board (two real, distinct roots). Zero = one passenger boards and stays seated (one repeated root). Negative = NO ONE boards — the jeepney is empty (complex/imaginary roots, no real solution). The Philippine jeepney is instantly relatable, and 'how many passengers' maps perfectly to 'how many real roots.'

Anchor Type

analogy

Why It Works

Concrete analogies anchor abstract mathematical concepts to vivid sensory experiences. The Filipino jeepney is an emotionally charged, culturally specific image that triggers instant recognition.

Example Usage

For x² + 4x + 5 = 0: discriminant = 16 − 20 = −4. Negative → 'jeepney is empty' → complex roots. Do not attempt to find real solutions.

Recall Trigger

Visualize a jeepney at a stop — how many passengers board?

Tags

• formula
• roots
• Vieta's formulas
• quadratic

Topic

Quadratic Equations

Concept

Sum of roots = −b/a; Product of roots = c/a

Anchor Id

A3

Difficulty

easy

Memory Aid

Remember 'SPC': Sum → negative b over a (S = −B/A), Product → c over a (P = C/A). Or use the phrase 'Suma Pababa, Produkto Pataas' — Sum comes from the top-left coefficient (−b/a, you negate b), Product comes from the bottom ratio (c/a, no negation). The Tagalog phrase 'Suma Pababa, Produkto Pataas' means 'Sum downward (negative), Product straight up (positive c/a).'

Anchor Type

acronym

Why It Works

Bilingual mnemonics are highly effective for Filipino students because they use the dominant language of daily thought. The directional metaphor (pababa = going down/negative, pataas = going up/positive) adds a spatial dimension to the memory.

Example Usage

For 3x² − 7x + 2 = 0: Sum = −(−7)/3 = 7/3; Product = 2/3. Verify: roots are x = 2 and x = 1/3. Sum = 2 + 1/3 = 7/3 ✓; Product = 2 × 1/3 = 2/3 ✓.

Recall Trigger

Say 'Suma Pababa, Produkto Pataas' in your head

Tags

• formula
• AP
• arithmetic progression
• sequence

Topic

Progressions

Concept

Arithmetic Progression (AP): aₙ = a₁ + (n−1)d

Anchor Id

A4

Difficulty

easy

Memory Aid

Imagine an OFW (Overseas Filipino Worker) named Ariana Progression who starts her savings at ₱a₁ on day 1 and adds exactly ₱d (daily allowance) every single day. On day n, her total savings-per-day rate is a₁ + (n−1)d — she has received the daily allowance (n−1) times after the first day. The 'minus 1' is because she started with a₁ on Day 1, not Day 0 — she didn't receive an allowance before arriving.

Anchor Type

micro_story

Why It Works

Micro-stories embed formula components into a narrative with logical cause and effect. The OFW context is deeply meaningful to Filipino students, creating emotional engagement that dramatically aids retention.

Example Usage

Find the 15th term of AP: 5, 9, 13, ... Here a₁ = 5, d = 4. a₁₅ = 5 + (15−1)(4) = 5 + 56 = 61.

Recall Trigger

Visualize Ariana Progression counting her daily savings abroad

Tags

• formula
• AP
• sum
• series

Topic

Progressions

Concept

AP Sum Formula: Sₙ = n/2 × (a₁ + aₙ) or n/2 × [2a₁ + (n−1)d]

Anchor Id

A5

Difficulty

easy

Memory Aid

The AP sum formula is like computing the AVERAGE SALARY of a department with n employees: (first salary + last salary) / 2 is the average, then multiply by the number of employees n. Sₙ = n × average = n × (a₁ + aₙ)/2. Gauss discovered this as a child by pairing numbers at opposite ends of 1 to 100 — each pair sums to 101, and there are 50 pairs: 50 × 101 = 5050. Same idea: pair the first and last, multiply by n/2.

Anchor Type

analogy

Why It Works

The payroll analogy connects to real-world financial thinking that engineering graduates understand. The Gauss story adds historical narrative that makes the formula's origin memorable.

Example Usage

Sum of first 10 terms: a₁ = 3, d = 4. a₁₀ = 3 + 9(4) = 39. S₁₀ = 10/2 × (3 + 39) = 5 × 42 = 210.

Recall Trigger

Think of 'average salary of a department' — first plus last, divided by 2, times number of people

Tags

• formula
• GP
• infinite series
• convergence

Topic

Progressions

Concept

Geometric Progression infinite sum: S∞ = a₁/(1−r), valid only for |r| < 1

Anchor Id

A6

Difficulty

medium

Memory Aid

Imagine a 'Forever Kuripot' (infinitely stingy) boss named Geo Metro who promises a₁ pesos salary today, then only pays a fraction r of last month's salary every month — forever. If r < 1, the total he EVER pays is a₁/(1−r). If r ≥ 1, the salary keeps growing or stays the same, and the total becomes INFINITE — Geo Metro goes bankrupt! The |r| < 1 condition is literally the 'convergence condition' — the boss must keep paying LESS each time.

Anchor Type

micro_story

Why It Works

The salary story makes the convergence condition logical and memorable. 'Geo Metro' phonetically echoes 'Geometric' for instant recall. The bankruptcy consequence makes the r ≥ 1 case unforgettable.

Example Usage

GP: 8, 4, 2, ... → a₁ = 8, r = 1/2. |r| = 1/2 < 1 ✓. S∞ = 8/(1 − 1/2) = 8/(1/2) = 16.

Recall Trigger

Remember 'Geo Metro, the forever kuripot boss' — salary shrinks each month

Tags

• work problems
• rates
• combined work
• word problems

Topic

Work Problems

Concept

Work Problems: Add RATES, not times. Rate = 1/time

Anchor Id

A7

Difficulty

medium

Memory Aid

Work problems are like ADDING TRICYCLE SPEEDS on EDSA — you add how fast each vehicle covers distance (rate), not how long the whole trip takes. If A does a job in 6 days and B in 4 days, their rates are 1/6 and 1/4 job/day. Combined rate = 1/6 + 1/4 = 5/12 job/day. Time together = 12/5 = 2.4 days. NEVER add '6 + 4 = 10 days' — that's like saying two tricycles traveling together take longer!

Anchor Type

analogy

Why It Works

The common mistake (adding times) is explicitly contrasted with the correct method (adding rates), which psychologists call 'error contrast learning.' The EDSA traffic reference is viscerally familiar to Metro Manila students.

Example Usage

Pipe A fills in 3h, Pipe B in 5h, Drain C empties in 4h. Net rate = 1/3 + 1/5 − 1/4 = 20/60 + 12/60 − 15/60 = 17/60 tank/hour. Time = 60/17 ≈ 3.53 hours.

Recall Trigger

Think 'add speeds on EDSA, not travel times' — add rates, not times

Tags

• logarithm
• common mistake
• log rules
• pitfall

Topic

Logarithms

Concept

Log of a sum ≠ sum of logs: log(M + N) ≠ log M + log N

Anchor Id

A8

Difficulty

medium

Memory Aid

Visualize a RED STOP SIGN shaped like a PLUS SIGN (+) inside a logarithm. The stop sign means: STOP — you CANNOT split a log over addition. log(M + N) is a TRAP. The only valid log rules split MULTIPLICATION (log MN = log M + log N) and DIVISION (log M/N = log M − log N) and POWERS (log Mᵖ = p log M). Draw this stop sign in your mind every time you see log(something + something).

Anchor Type

visual_association

Why It Works

Visual warning associations (stop sign = danger) tap into procedural safety memory — a deeply ingrained reflex. This is one of the most commonly tested traps in PRC board exams.

Example Usage

If log 2 = 0.301 and log 3 = 0.477, find log 5. Note: 5 = 10/2, so log 5 = log 10 − log 2 = 1 − 0.301 = 0.699. Do NOT try log(2 + 3) = log 5 via addition.

Recall Trigger

See a red STOP SIGN whenever you see log(M + N)

Tags

• logarithm
• log rules
• formula
• properties

Topic

Logarithms

Concept

Log rules: Product → Sum, Quotient → Difference, Power → Multiply

Anchor Id

A9

Difficulty

easy

Memory Aid

Use the acronym 'PSQ-DPM': Products become Sums (log MN = log M + log N), Quotients become Differences (log M/N = log M − log N), Powers Multiply out front (log Mᵖ = p·log M). Or remember 'PSD': Product = Sum, Division = Difference, Power = front-multiply. Think of LOGARITHMS AS A TRANSLATOR: multiplication language → addition language. This is why engineers use log tables — they convert hard multiplication into easy addition.

Anchor Type

acronym

Why It Works

The 'translator' analogy gives logs a purpose and a persona, making the rules feel logical rather than arbitrary. The PSD acronym provides quick recall under exam pressure.

Example Usage

Simplify log 4 + log 25. = log(4 × 25) = log 100 = 2. Or: log 8 = log 2³ = 3 log 2 = 3(0.301) = 0.903.

Recall Trigger

Log is a TRANSLATOR: PSD — Products→Sums, Division→Difference, Power→front-multiply

Tags

• logarithm
• change of base
• formula
• calculator technique

Topic

Logarithms

Concept

Change of Base Formula: log_b(x) = ln x / ln b

Anchor Id

A10

Difficulty

medium

Memory Aid

Change of base is like converting Philippine Pesos to US Dollars — you need an exchange rate. The 'exchange rate' is ln b (or log b). To express log base b of x, you divide ln x by ln b. Your calculator speaks only ln or log₁₀ — change of base is your CURRENCY CONVERTER that translates any exotic base to one your calculator understands.

Anchor Type

analogy

Why It Works

Currency conversion is a daily reality for Filipino engineers, especially those with OFW family members. The concrete financial analogy makes the abstract formula operationally intuitive.

Example Usage

Evaluate log₅(125). log₅(125) = ln 125 / ln 5 = 4.828 / 1.609 = 3. Or recognize 5³ = 125, so log₅(125) = 3.

Recall Trigger

Think 'currency converter — divide by the exchange rate (ln b)'

Tags

• exponents
• laws of exponents
• formula
• basic algebra

Topic

Exponents and Radicals

Concept

Laws of Exponents: aᵐ · aⁿ = aᵐ⁺ⁿ (same base, add exponents)

Anchor Id

A11

Difficulty

easy

Memory Aid

Engineer Arnel has a sack with 'a' multiplied by itself m times, then another sack with 'a' multiplied n times. When he COMBINES them, he has aᵐ × aⁿ = aᵐ⁺ⁿ — he just COUNTS all the a's together. He remembers: 'SAME BASE, ADD POWERS.' But when bases differ (e.g., 2³ × 3²), you CANNOT add exponents — they're like different currencies, you can't just add pesos and dollars.

Anchor Type

micro_story

Why It Works

The sack-counting image gives a physical, tactile grounding for an abstract rule. The 'different currencies' warning for different bases preemptively addresses the most common error.

Example Usage

Simplify 2⁴ · 2³ = 2⁴⁺³ = 2⁷ = 128. Also: (x³)⁴ = x¹² (multiply exponents when raising a power to a power).

Recall Trigger

Visualize Arnel counting items in combined sacks — same base, count all exponents

Tags

• binomial theorem
• formula
• expansion
• specific term

Topic

Binomial Theorem

Concept

Binomial Theorem: (r+1)th term = C(n,r) · aⁿ⁻ʳ · bʳ

Anchor Id

A12

Difficulty

hard

Memory Aid

The specific term in binomial expansion follows the 'NBA JERSEY RULE': Number (C(n,r)) × A-power (aⁿ⁻ʳ) × B-power (bʳ). The r in the jersey number = r in bʳ (b's exponent equals r). The (r+1)th term has r starting at 0, so the 6th term has r = 5. Remember: 'JERSEY NUMBER STARTS AT ZERO — rookie r=0 is the first term, veteran r=5 is the 6th term.'

Anchor Type

visual_association

Why It Works

Basketball jersey numbers are familiar to young Filipino engineers. The 'r starts at 0' confusion is one of the most common board exam errors; anchoring it to a familiar numbering system (jerseys start at 0 for the 1st player) helps prevent off-by-one errors.

Example Usage

Find the 6th term of (2x − 1)⁹. Here n=9, r=5. T₆ = C(9,5)·(2x)⁴·(−1)⁵ = 126 · 16x⁴ · (−1) = −2016x⁴.

Recall Trigger

Think 'NBA Jersey Rule — r=0 means Term 1, r=5 means Term 6'

Tags

• mixture problems
• word problems
• concentration
• equation setup

Topic

Mixture Problems

Concept

Mixture Problems: Set up equations tracking QUANTITY × CONCENTRATION = AMOUNT of substance

Anchor Id

A13

Difficulty

medium

Memory Aid

Mixture problems are like making SINIGANG with different sourness levels. If you mix 3 liters of mild broth (20% sourness) with 2 liters of super sour broth (50% sourness), the total sour content is 3(0.20) + 2(0.50) = 1.6 liters. Total volume is 5 liters. Final concentration = 1.6/5 = 32%. The key: TRACK THE SOUR STUFF, not the total broth. Always set up: (volume₁)(conc₁) + (volume₂)(conc₂) = (total volume)(final conc).

Anchor Type

analogy

Why It Works

Sinigang is intimately familiar to every Filipino student. Mapping 'sourness concentration' to mathematical concentration makes the setup equation intuitive and culturally resonant.

Example Usage

10 L of 30% acid + x L of 60% acid = 20 L of y% acid. Equation: 10(0.30) + x(0.60) = 20(y). Also, 10 + x = 20, so x = 10. Then 3 + 6 = 20y → y = 45%.

Recall Trigger

Making sinigang — track the sour substance, not the total broth

Tags

• age problems
• word problems
• equation setup
• algebra

Topic

Age Problems

Concept

Age Problems: Represent ages in terms of ONE variable; set up equations for 'years ago' and 'years hence'

Anchor Id

A14

Difficulty

medium

Memory Aid

Picture an exam proctor named Lolo Algebra who is 3 times as old as his apo (grandchild). 'Years ago' means SUBTRACT from BOTH ages; 'years hence' means ADD to BOTH ages. Many board examinees fail age problems because they only change ONE person's age. Remember: TIME FLOWS FOR EVERYONE. If 10 years ago Lolo was 3× the apo's age, write: (L − 10) = 3(A − 10). Lolo's age went back 10, the apo's age went back 10 — both move through time together.

Anchor Type

micro_story

Why It Works

The grandfather-grandchild (lolo-apo) relationship is universally recognized in Filipino culture, creating an emotionally warm, memorable context. 'Time flows for everyone' is a memorable phrase that prevents the common error.

Example Usage

Ana is 3× as old as Ben. 8 years ago, Ana was 5× Ben's age. Let B = Ben's current age. 3B − 8 = 5(B − 8) → 3B − 8 = 5B − 40 → 2B = 32 → B = 16. Ana = 48.

Recall Trigger

Lolo Algebra and his apo — time flows for EVERYONE, subtract/add from BOTH ages

Tags

• motion problems
• DRT
• distance rate time
• word problems

Topic

Motion Problems

Concept

Distance = Rate × Time (DRT) and the two classic setups: opposite directions and same direction

Anchor Id

A15

Difficulty

medium

Memory Aid

DRT is the 'TRIANGLE OF MOTION' — D on top, R and T on the bottom corners. Cover what you want: cover D to get R×T; cover R to get D/T; cover T to get D/R. For OPPOSITE DIRECTIONS (approaching): add speeds, since they close the gap together: D = (r₁ + r₂) × t. For SAME DIRECTION (chasing): subtract speeds, since the faster one gains ground: D = (r₁ − r₂) × t. Think: 'Opposite = Magkasalungat = ADD; Same direction = Magkasabay = SUBTRACT.'

Anchor Type

visual_association

Why It Works

The triangle cover method is a classic spatial mnemonic that allows formula rearrangement without algebra. The Tagalog directional words (magkasalungat = opposing, magkasabay = in the same direction) create a bilingual hook.

Example Usage

Two trains start 300 km apart, approaching each other at 60 and 90 km/h. Time to meet: 300 = (60 + 90)t → t = 2 hours. If chasing: faster at 90, slower at 60, 30 km apart. Time: 30 = (90 − 60)t → t = 1 hour.

Recall Trigger

DRT triangle — opposite means ADD speeds, same direction means SUBTRACT

Tags

• formula
• GP
• geometric progression
• sum

Topic

Progressions

Concept

GP Sum Formula: Sₙ = a₁(rⁿ − 1) / (r − 1) for r ≠ 1

Anchor Id

A16

Difficulty

medium

Memory Aid

Remember the LEGEND OF THE RICE GRAINS CHESS BOARD — an ancient inventor asked a king for rice: 1 grain on square 1, 2 grains on square 2, 4 grains on square 3, doubling every square. The total is S₆₄ = 1(2⁶⁴ − 1)/(2 − 1) = 2⁶⁴ − 1 ≈ 18 quintillion grains. The king went bankrupt. The formula: a₁ = 1, r = 2, n = 64. The parenthetical '(rⁿ − 1)' in the numerator is the 'ever-growing total minus the starting point'; dividing by '(r − 1)' is the 'scaling back to reality.'

Anchor Type

micro_story

Why It Works

The chess board story is one of the world's most famous mathematical stories, demonstrating the power of geometric growth. The bankruptcy consequence (like Geo Metro in A6) makes the formula emotionally vivid.

Example Usage

GP: 3, 6, 12, 24, ... Find S₈. a₁ = 3, r = 2, n = 8. S₈ = 3(2⁸ − 1)/(2 − 1) = 3(255)/1 = 765.

Recall Trigger

Chess board rice grains — a₁(rⁿ − 1) / (r − 1)

Tags

• exponents
• zero exponent
• negative exponent
• laws of exponents

Topic

Exponents and Radicals

Concept

a⁰ = 1 for any non-zero a, and a⁻ⁿ = 1/aⁿ

Anchor Id

A17

Difficulty

easy

Memory Aid

Think of a⁰ as 'empty hands — you haven't multiplied by anything yet, so you start with the identity: 1.' A negative exponent means you OWE multiplication — a⁻ⁿ means you owe n multiplications, so it becomes 1/aⁿ (it flips to the denominator). Like a UTANG (debt): negative exponent = your multiplication is on the other side of the fraction bar, paying off the debt.

Anchor Type

analogy

Why It Works

The 'utang' (debt) metaphor is culturally resonant and makes the 'flip to denominator' rule logical rather than arbitrary. Empty hands for zero exponent gives a physical, intuitive grounding.

Example Usage

5⁰ = 1. 2⁻³ = 1/2³ = 1/8. (3x)⁻² = 1/(3x)² = 1/(9x²). Always flip negative exponents before computing.

Recall Trigger

Negative exponent = UTANG — flip it to the denominator to pay the debt

Tags

• radicals
• fractional exponents
• formula
• conversion

Topic

Exponents and Radicals

Concept

Radical as fractional exponent: √[n]{aᵐ} = aᵐ/ⁿ

Anchor Id

A18

Difficulty

easy

Memory Aid

The radical symbol √ is shaped like a DIVISION SIGN inside a roof. The number inside the radical (m) goes to the TOP of the fraction exponent (numerator = m); the index of the radical (n, the small number outside) goes to the BOTTOM (denominator = n). So √[n]{aᵐ} = aᵐ/ⁿ. Visualize the radical as a HOUSE: the power (m) lives inside the house (numerator), the index (n) is the house number on the bottom (denominator).

Anchor Type

visual_association

Why It Works

Spatial mapping (inside = numerator, outside = denominator) exploits the brain's visual-spatial processing. The house metaphor gives the two positions a memorable logical structure.

Example Usage

Convert ∛(x²) to exponential form: n=3 (index), m=2 (power). x²/³. Also: 8^(2/3) = (∛8)² = 2² = 4.

Recall Trigger

House metaphor — m is INSIDE (numerator), n is the house NUMBER outside (denominator)

Tags

• binomial coefficient
• combination
• formula
• binomial theorem

Topic

Binomial Theorem

Concept

Binomial Coefficient C(n,r) = n! / [r!(n−r)!]

Anchor Id

A19

Difficulty

medium

Memory Aid

Remember 'CeNteR' — C is top (n), bottom is r and the REST (n−r). C(n,r) = n! / [r! × (n−r)!]. The 'center' trick: n is the total, r is what you CHOOSE, (n−r) is what you LEAVE. Think of choosing a 5-person team from 9 engineers: C(9,5) = 9! / (5! × 4!). The two bottom factorials always add up to the top: r + (n−r) = n. If they don't add up, you made an error — this is your built-in CHECK.

Anchor Type

mnemonic

Why It Works

The 'choose and leave' framing makes the formula's structure logical. The check (r + (n−r) = n) provides a self-verification mechanism that reduces errors under exam pressure.

Example Usage

C(9,5) = 9! / (5! × 4!) = (9×8×7×6)/(4×3×2×1) = 3024/24 = 126. Check: 5 + 4 = 9 ✓.

Recall Trigger

CeNteR: n on top, r and (n−r) on bottom — choose and leave, they must sum to n

Tags

• logarithm
• common mistake
• pitfall
• board trap

Topic

Logarithms

Concept

Common board trap: log(M+N) ≠ log M + log N; only PRODUCTS split into sums

Anchor Id

A20

Difficulty

medium

Memory Aid

Rookie engineer Ernie (ERNIE = ERRor for Novice in Exams) always fails by writing 'log(M+N) = log M + log N' on his boards. He got this from confusing log rules with exponent rules. His mentor, senior engineer Beling, told him: 'Ernie, magmultiply ka lang, hindi magdagdag!' (Ernie, only multiply, don't add!) The LOG SPLITS MULTIPLICATION into addition — NEVER addition into addition. Ernie failed his first board. Don't be Ernie.

Anchor Type

micro_story

Why It Works

Failure stories activate the brain's loss-aversion system, making warnings more memorable than positive examples. The Tagalog phrase from a mentor figure adds cultural warmth and authority.

Example Usage

log(100) ≠ log(60 + 40) ≠ log 60 + log 40. But log(100) = log(4 × 25) = log 4 + log 25 = 2 ✓.

Recall Trigger

Don't be Ernie — log splits only MULTIPLICATION (×) into addition (+)

Formula

x = (−b ± √(b²−4ac)) / 2a

Mnemonic

Sing to 'Pop Goes the Weasel': 'x equals NEGATIVE b, PLUS OR MINUS square root, of B-SQUARED minus FOUR-A-C, ALL OVER TWO-A!' The musical sequence locks in the exact order: negative sign on b, ±, square root, b², minus, 4ac, denominator 2a.

When To Use

When factoring is not obvious or not possible. Whenever the quadratic ax² + bx + c = 0 needs to be solved for x. Also used to derive the discriminant for nature-of-roots problems.

What Each Part Means

a = coefficient of x², b = coefficient of x, c = constant term. The ± gives two roots. b²−4ac is the discriminant (nature of roots indicator). 2a is the denominator scaling factor.

Formula

aₙ = a₁ + (n−1)d

Mnemonic

OFW ARIANA: 'Ariana starts at a₁, gets allowance d each of (n−1) days after arrival.' The (n−1) comes because she starts WITH money on Day 1 — the allowance only kicks in on subsequent days. Phrase: 'Start Plus (Steps Times Difference)' — S + (n−1)d.

When To Use

To find any specific term in an arithmetic progression. Also used to find n when the nth term is given.

What Each Part Means

a₁ = first term, d = common difference (positive = increasing AP, negative = decreasing AP), n = term number, aₙ = nth term.

Formula

Sₙ = n/2 × (a₁ + aₙ) = n/2 × [2a₁ + (n−1)d]

Mnemonic

GAUSS PAYROLL: 'Average of first and last salary × number of employees.' Sₙ = n × [(a₁ + aₙ)/2]. The two forms are equivalent — use the first when both a₁ and aₙ are known; use the second when only a₁ and d are known.

When To Use

To find the sum of a finite arithmetic series. Board problems often give a₁, d, and n, requiring S; or give S and ask for n or d.

What Each Part Means

n = number of terms, a₁ = first term, aₙ = last term, d = common difference. The n/2 factor represents the average pairing of terms.

Formula

aₙ = a₁ × r^(n−1)

Mnemonic

GP CHESS BOARD: 'Start with a₁ grain on square 1, multiply by r exactly (n−1) times to reach square n.' The (n−1) is parallel to AP — you multiply r zero times at term 1 (just a₁), once at term 2 (a₁r), and so on.

When To Use

To find any specific term in a geometric progression. Also used to find r or a₁ when other terms are given.

What Each Part Means

a₁ = first term, r = common ratio (r > 1 growing, 0 < r < 1 decaying, r < 0 alternating), n = term number.

Formula

Sₙ = a₁(rⁿ − 1) / (r − 1)

Mnemonic

RICE GRAINS FORMULA: 'Total grains = (starting grain)(how much r grows to the nth power, minus the start) divided by (the growth rate minus 1).' Alternative form: a₁(1 − rⁿ)/(1 − r) for |r| < 1. Both are equivalent — use whichever keeps the signs positive.

When To Use

To find the sum of a finite geometric series with r ≠ 1. If r = 1, Sₙ = n × a₁ (trivial case — add n identical terms).

What Each Part Means

a₁ = first term, r = common ratio (r ≠ 1), n = number of terms. The numerator (rⁿ − 1) represents the total geometric growth. The denominator (r − 1) normalizes back to the original scale.

Formula

S∞ = a₁ / (1 − r), valid for |r| < 1

Mnemonic

GEO METRO'S SALARY: 'Boss pays forever, shrinking every time. Total ever paid = first payment / (1 minus the shrink rate).' The (1 − r) in the denominator represents how much room there is for the series to converge before hitting 1. If r = 0.5, there's 0.5 room left → S∞ = a₁/0.5 = 2a₁.

When To Use

To find the sum of an infinitely repeating geometric series. Classic board applications: repeating decimals (0.333... = 3/10 + 3/100 + ... = (3/10)/(1 − 1/10) = 1/3) and infinite bouncing ball problems.

What Each Part Means

a₁ = first term, r = common ratio with |r| < 1 (mandatory convergence condition). The formula fails (S∞ → ∞) if |r| ≥ 1.

Formula

(r+1)th term of (a+b)ⁿ = C(n,r) × aⁿ⁻ʳ × bʳ

Mnemonic

NBA JERSEY RULE: 'Rookie r=0 wears Jersey 1 (first term). Veteran r=5 wears Jersey 6 (sixth term). Term number = r + 1, so r = term number − 1.' Components: C(n,r) is the coefficient, aⁿ⁻ʳ has a's exponent going DOWN from n, bʳ has b's exponent going UP from 0. Sum of exponents always = n: (n−r) + r = n.

When To Use

To find a specific term in a binomial expansion without expanding the entire expression. Board problems often ask for 'the middle term,' 'the 5th term,' or 'the term containing x³.'

What Each Part Means

n = power of binomial, r = zero-based index of the term (r = 0 for first term), C(n,r) = binomial coefficient, a and b are the two terms of the binomial.

Formula

log_b(x) = ln x / ln b (Change of Base)

Mnemonic

CURRENCY CONVERTER: 'Convert foreign log to calculator log by dividing by the exchange rate.' log_b(x) = log x / log b = ln x / ln b. The denominator is ALWAYS the original base b. Trick: if you see log₅(125), change to ln 125 / ln 5 or log 125 / log 5 = 2.097 / 0.699 = 3.

When To Use

Whenever the base is not 10 or e and you need to evaluate on a calculator. Also used to prove identities involving different bases.

What Each Part Means

log_b(x) = log of x in base b (the 'foreign currency'). ln x = natural log of x (your calculator's language). ln b = natural log of the base (the exchange rate). You can also use log₁₀ instead of ln — the ratio is the same.

Chain Title

Laws of Exponents — The 6 Rules

Recall Test

Without looking: What is the rule for (a³)⁴? For a⁻²? For (2x)³?

Memory Chain

Arnel the Engineer follows the 'SASP-ZF' rule: Same base Add (×), Same base Subtract (÷), Stack powers Multiply (power of power), Spread over product, Zero=1, Flip negative. Chain story: 'Arnel adds his sacks of a (aᵐ⁺ⁿ), subtracts when sharing (aᵐ⁻ⁿ), multiplies when stacking crates (aᵐⁿ), spreads seeds over all pots ((ab)ⁿ), has nothing (zero=1 because he still has the empty basket), then goes into debt (negative flips).'

Items To Remember

• aᵐ · aⁿ = aᵐ⁺ⁿ (same base, add)
• aᵐ / aⁿ = aᵐ⁻ⁿ (same base, subtract)
• (aᵐ)ⁿ = aᵐⁿ (power of a power, multiply)
• (ab)ⁿ = aⁿbⁿ (distribute over product)
• a⁰ = 1 (zero exponent is one)
• a⁻ⁿ = 1/aⁿ (negative flips to denominator)

Chain Title

Log Rules — The 4 Properties

Recall Test

Simplify without calculator: log₂(32) + log₂(2). What is log₅(1)? What is log₃(3⁵)?

Memory Chain

Remember 'PSDBI': Product=Sum, Division=Difference, Big power → little multiplier in front, Base of self=1, Identity 1=zero. Story: 'The LOG TRANSLATOR: multiplication becomes addition (× → +), division becomes subtraction (÷ → −), powers become multipliers (front). Special IDs: log of your own base is 1 (log₂2=1), log of 1 is always 0 (log 1=0).'

Items To Remember

• log(MN) = log M + log N (product → sum)
• log(M/N) = log M − log N (quotient → difference)
• log(Mᵖ) = p · log M (power → multiply front)
• log_b(b) = 1 and log_b(1) = 0 (base identities)

Chain Title

Work Problem Setup — 5-Step Method

Recall Test

A takes 8h alone, B takes 12h alone. Without calculating: what is the formula setup for their combined time? What if there's also a drain that empties in 6h?

Memory Chain

TARST Chain — 'Tara, Add Rates, Solve T!' Time alone → Rate (invert) → Add all rates → Set equal to 1/T → Solve T. Micro-story: 'Tara the plumber (T) Always Runs Straight to the finish — she doesn't add times, she adds rates!'

Items To Remember

• Step 1: Identify each worker's time to finish ALONE
• Step 2: Convert each time to a RATE (1/time)
• Step 3: ADD rates (or subtract for drain/opponent)
• Step 4: Combined rate = 1/T (where T is time together)
• Step 5: Solve for T = 1 / combined rate

Chain Title

Discriminant and Nature of Roots

Recall Test

For x² + 6x + 9 = 0: compute D. What kind of roots? For x² + x + 1 = 0: compute D. Any real solutions?

Memory Chain

JEEPNEY CHAIN: 'D is the jeepney's passenger count formula. Compute b²−4ac. Positive D → 2 passengers board (2 real roots). Zero D → 1 passenger boards and stays (1 repeated root). Negative D → jeepney is empty, no boarding (complex roots).' Sequence: compute D → check sign → declare root type.

Items To Remember

• D = b² − 4ac (the discriminant formula)
• D > 0: two distinct real roots
• D = 0: one repeated real root (roots are equal)
• D < 0: two complex conjugate roots (no real solution)

Chain Title

Binomial Theorem — Steps to Find the rth Term

Recall Test

Find the 4th term of (x + 2)⁷. What is r? What is C(7,r)? What powers do x and 2 carry?

Memory Chain

NBA JERSEY CHAIN: 'To find the 6th term: Rookie r = 6 − 1 = 5 (veteran jersey). Combo C(n,5). First term power = n − 5 (going down). Second term power = 5 (going up). Multiply: C(n,5) × a^(n-5) × b^5. Remember: powers must SUM to n: (n−5)+5=n ✓.'

Items To Remember

• Step 1: Identify term number T (e.g., 5th term)
• Step 2: Find r = T − 1 (since term = r+1, so r = T−1)
• Step 3: Write C(n, r) as coefficient
• Step 4: First term raised to (n − r)
• Step 5: Second term raised to r
• Step 6: Multiply all three parts including sign