high school Geometric Sequence

Geometric Sequence

A geometric sequence is built by repeatedly multiplying the previous term by the same number. When the common ratio r is above 1 the terms explode like a snowball, between 0 and 1 they fade toward zero, and when r is negative the signs alternate. Any term is first term × r^(n − 1), and the sum folds into one formula; remarkably, when |r| is below 1 even an endless sum settles on a single value. Change the first term and ratio and the terms grow or fade, and an infinite series converges or diverges.

Geometric Sequence — Multiply by the Same Ratio

First term a₁2

Common ratio r2

Number of terms n5

💡 Geometric = sequence with constant ratio

①Adjacent terms share the same ratio (= r)

②r > 1: exponential growth

③0 < r < 1: exponential decay

④r < 0: signs alternate

⑤The first-row ratio is r ≥ 0.3. r < 0 is the convergence slider

nth Term & Sum

nth Term

a_n = a₁ · r^n-1

nth term = first term × r⁽n-1)

Sum

S_n = a₁(rⁿ - 1)r - 1 (r ≠ 1)

If r = 1, S_n = na₁

📐 Deriving the Sum

①S_n = a₁ + a₁r + a₁r² + ... + a₁r^{n-1}

②rS_n = a₁r + a₁r² + ... + a₁r^n

③Subtract: S_n − rS_n = a₁ − a₁r^n

④So S_n(1−r) = a₁(1−r^n)

Geometric Series — Convergence

Ratio r0.5

Infinite Geometric Series

S = a₁1 - r (|r| < 1)

Converges only when |r| < 1; diverges otherwise. This infinite-sum canvas uses a₁ = 1

Geometric Mean

b² = ac (a, b, c geometric)

Square of the middle = product of the outer terms

📐 Using the Geometric Mean

①Three numbers a, b, c geometric ⟺ b² = ac

②b = ±√(ac) — watch the sign

③Equally-spaced subseries are also geometric

Wrap-up

Geometric Core

a_n = a₁r^n-1, S_n = a₁(rⁿ - 1)r - 1

nth term and sum in one place

🎯 Exam Points

①nth term: a_n = a₁ · r^{n-1}

②Sum: S_n = a₁(r^n − 1)/(r − 1) (r ≠ 1)

③Infinite series: |r| < 1 gives S = a₁/(1−r)

④Geometric mean: b² = ac

⑤Sign and magnitude of r govern the behavior

Worked Examples & Past Exam

Example 1

Find the 5th term of a geometric sequence with first term 3 and common ratio 2.

Substitute into the general term a_n = a₁ · r^n-1.

a₅ = 3 × 2^5-1 = 3 × 2⁴

Compute.

= 3 × 16 = 48

▸ 5th term = 48

The general term multiplies the first term by the ratio (n−1) times.

Example 2

Find the sum of the first 4 terms of a geometric sequence with first term 2 and common ratio 3.

Substitute into the sum formula S_n = a₁(rⁿ − 1)/(r − 1).

S₄ = 2(3⁴ - 1)3 - 1

Compute.

= 2(81 - 1)2 = 80

▸ sum = 80

The sum of a geometric sequence is a₁(rⁿ − 1)/(r − 1) for r ≠ 1.

2022 KICE mock exam Math type, adapted

Find the sum of the infinite geometric series with first term 4 and common ratio 1/2.

①8

②4

③2

④6

⑤does not converge

▸ ① 8

Since |r| = 1/2 < 1 it converges; use S = a₁/(1 − r).

S = a₁1 - r = 41 - 1/2

Compute.

= 41/2 = 8

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Arithmetic Sequence

Sum of Series

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