Overview

Geometric sequences multiply by a constant ratio. These are the foundations of exponential growth and infinite series, shown in many AMC Problems.

Key Ideas

• $g_n = g_1 r^{n-1}$.
• For finite sums, $S_n = g_1(1-r^n)/(1-r)$. For infinite sums, one must find that $|r| < 1$.
• Infinite sums converge only when $|r| < 1$.

The General Term

For a geometric sequence with first term $g_1$ and common ratio $r$,

$g_n = g_1 \cdot r^{n-1}.$

The ratio is always $r = g_{k+1}/g_k$ for any consecutive pair.

For non-consecutive terms $g_m$ and $g_n$,

$r^{n-m} = \frac{g_n}{g_m} \implies r = \left(\frac{g_n}{g_m}\right)^{\!\frac{1}{n-m}}.$

The Finite Sum

$S_n = g_1 \cdot \frac{1-r^n}{1-r}, \quad r \ne 1.$

When $r = 1$: $S_n = n \cdot g_1$. We can derive this by multiplying the sum by $r$, and subtracting. Most terms cancel out, and we obtain $S(1-r) = g_1(1-r^n)$.

The Infinite Sum

When $|r| < 1$ the terms shrink to zero and the series converges:

$S_\infty = \frac{g_1}{1-r}, \quad |r| < 1.$

When $|r| \geq 1$ the series diverges, and thus cannot apply this formula.

Geometric Mean

Three numbers $a, b, c$ are in geometric progression if and only if $b^2 = ac$. The number $b$ is the geometric mean of $a$ and $c$.

Hence for positive numbers, the geometric mean of $a$ and $c$ is $\sqrt{ac}$.

Symmetric Substitution

For problems giving the sum and product of three terms in GP, we write them as $a/r,\ a,\ ar$.

The product is shown through $a^3$, so $a$ can be found. Thus, we know that the sum gives a quadratic in $r$.

Worked Example

Find $1 + \frac{1}{3} + \frac{1}{9} + \cdots$.

This is geometric with $g_1 = 1$ and $r = \frac{1}{3}$. Since $|r| < 1$, $S_\infty = \frac{1}{1-\frac{1}{3}} = \frac{3}{2}$.

More Examples

Example 1: Using the Term Formula, Basic

If $g_1=6$ and $r=2$, find $g_5$.

$g_5 = 6\cdot 2^4 = 96$.

Example 2: Finite Sum Equation, (Must memorize for Foundational Skill)

Find the sum of the first 6 terms of $3,6,12,\ldots$.

$S_6 = 3\cdot (1-2^6)/(1-2) = 3(63) = 189$.

Example 3: Using Convergence

Does $5 - 2.5 + 1.25 - \cdots$ converge? If so, find the sum.

$r=-1/2$ so $|r|<1$ and $S_\infty = 5/(1+1/2)=10/3$.

Example 4: Using Geometric Roots and applying to Vieta's Formulas

The roots of $x^3 - 14x^2 + 56x - 64 = 0$ are in GP, find these roots.

Let roots be $a/r,\ a,\ ar$. By Vieta's formulas, we know that the product: $a^3 = 64 \implies a = 4$, and the sum is equivalent to $4(1/r + 1 + r) = 14 \implies r + 1/r = 5/2 \implies 2r^2 - 5r + 2 = 0 \implies r = 2$ or $r = 1/2$. Roots: $\mathbf{2, 4, 8}$

Strategy Checklist

• Compute the ratio from consecutive terms for accuracy.
• Decide whether the sum is finite or infinite at first.
• Check $|r|<1$ before using the infinite formula.
• Keep powers as $r^{n-1}$, for convenience and less error (Highly recommended in more complex problems)

Common Pitfalls

• A common mistake is applying the infinite sum formula when $|r| \ge 1$.
• Mixing up $r^n$ vs. $r^{n-1}$ in the term formula can lead to error.
• Using $r$ from non-consecutive terms without checking consistency.

Practice Problems