5.3 Sequences, Series & Logarithms

Sequences & Series on a Timed Engineering Paper

Sequences (AP/GP), series sums, and logarithms appear throughout FSc maths and are standard in PAF CAE-oriented banks. With maths often described as ~50 MCQs in ~25 minutes (treat as a commonly reported pattern, not a guaranteed official constant), closed-form recall beats listing terms.

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.

On the Exam

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.