5.3 Sequences, Series & Logarithms

Sequences & Series

Sequences (AP/GP), series sums, and logarithms are standard school-level mathematics. Closed-form recall is more efficient than listing many terms, but this guide does not assign these topics an official PAF count or timing.

Arithmetic Progression (AP)

An AP has constant common difference d.

Worked nth term. AP: 7, 10, 13, … Find a₂₀.
a = 7, d = 3 → a₂₀ = 7 + 19·3 = 7 + 57 = 64.

Worked sum. Sum of first 20 terms: S₂₀ = 20/2 · [2·7 + 19·3] = 10·[14 + 57] = 10·71 = 710. Or S₂₀ = 20/2 · (7 + 64) = 10·71 = 710.

Three terms in AP. Often written a − d, a, a + d so the middle is the average—useful in word problems.

Geometric Progression (GP)

A GP has constant common ratio r.

Worked. GP: 3, 6, 12, … Find a₈ and S₈.
r = 2, a₈ = 3·2⁷ = 3·128 = 384.
S₈ = 3(2⁸ − 1)/(2 − 1) = 3(256 − 1) = 765.

Infinite GP. 1 + 1/2 + 1/4 + 1/8 + … → a = 1, r = 1/2, |r| < 1 → S_∞ = 1/(1 − 1/2) = 2.

If |r| ≥ 1, the infinite sum does not converge to a finite value—classic distractor.

Inserting Means

n arithmetic means between a and b: divide (b − a) into (n + 1) equal parts so d = (b − a)/(n + 1).

Worked. Three AMs between 4 and 16: d = (16 − 4)/4 = 3 → means 7, 10, 13.

Geometric means between a and b: common ratio satisfies rⁿ⁺¹ = b/a.

Relation Between AM and GM

For positive a, b: AM = (a + b)/2, GM = √(ab), and AM ≥ GM, equality iff a = b. Occasional theory MCQ.

Logarithms: Definition and Laws

Definition: log_b a = c ⇔ bᶜ = a, with b > 0, b ≠ 1, a > 0.

Special values: log_b 1 = 0, log_b b = 1, log_b (bᵏ) = k.

Natural log ln x = log_e x; common log log x often means log₁₀ x in FSc contexts—read the stem.

Worked expansion. log(100√10) base 10: 100 = 10², √10 = 10^{1/2}, so log₁₀(10² · 10^{1/2}) = 2 + 1/2 = 2.5.

Worked change of base. log₂ 8 = ln 8 / ln 2 = 3 ln 2 / ln 2 = 3 (since 2³ = 8).

Worked equation. Solve log₁₀(x − 1) + log₁₀(x + 1) = log₁₀ 8.
Combine: log₁₀[(x − 1)(x + 1)] = log₁₀ 8 ⇒ x² − 1 = 8 ⇒ x² = 9 ⇒ x = ±3. Domain requires x − 1 > 0 and x + 1 > 0 for real logs of each factor (product form still needs x² − 1 > 0 and usually each positive depending on how the original was written). From the original sum of logs, need x > 1. So x = 3 only.

Exponential Equations

Worked. 3^{x+1} = 81. Note 81 = 3⁴, so 3^{x+1} = 3⁴ ⇒ x + 1 = 4 ⇒ x = 3.

Worked mixed. 2^{x} = 5 ⇒ x = log₂ 5 = ln 5 / ln 2 (leave in log form unless a decimal is offered).

Exponentials, Logs, and Graphs (Quick)

• y = aˣ (a > 1) increasing; y = aˣ (0 < a < 1) decreasing; horizontal asymptote y = 0.
• y = log_b x domain (0, ∞), vertical asymptote x = 0; passes through (1, 0) and (b, 1).

Series Notation

Σ from k = 1 to n of k = n(n + 1)/2.
Σ k² = n(n + 1)(2n + 1)/6.
Σ k³ = [n(n + 1)/2]².

Worked. Σ_{k=1}^{10} k = 10·11/2 = 55. Useful when an AP of natural numbers is disguised as sigma notation.

Practice Focus

Pattern-match AP vs GP in under five seconds (constant difference vs constant ratio). For logs, expand products/powers before substituting numbers. Reject negative or zero arguments for real logarithms even if algebra produces extra roots.