Geometric Progression on GMAT

50-Word Summary

A geometric progression is a sequence where each term is obtained by multiplying the previous term by a fixed common ratio. Example: 3, 6, 12, 24 has ratio 2. The nth-term is a·r^n-1, and the sum of n terms is Sₙ = a(rⁿ − 1)/(r − 1).

Overview

Geometric progressions are an important branch of sequences and series, and they often form the basis of elegant GMAT questions. A geometric progression is a sequence in which each term is obtained by multiplying the previous term by a constant ratio. Unlike arithmetic progressions, which grow by addition, geometric progressions grow (or shrink) by multiplication, creating patterns that expand quickly or diminish rapidly. What makes them particularly useful is that formulas exist for calculating the nth term, the sum of a fixed number of terms, and even the sum of infinitely many terms under specific conditions. These relationships allow us to handle large terms and complex-looking series without needing to write them out manually. Mastery of these formulas and the logic behind them saves time and reduces error. That is why understanding geometric progressions is an essential part of GMAT preparation, and reinforcing this knowledge through GMAT practice tests ensures strong performance under pressure.

 

Understanding Geometric Progressions on the GMAT

A geometric progression (GP) is a sequence of numbers where each term is obtained by multiplying the previous term by a constant ratio.

Examples of Geometric Progressions

• 1, 2, 4, 8… (common ratio = +2)
• 5, −20, 80… (common ratio = −4)
• 27, −9, 3, −1… (common ratio = −1/3)

Useful Relationships

The nth term of a Geometric Progression

tn = a × r ^{(n − 1)}
where a = first term, r = common ratio, and n = term position.

The sum of the first n terms of a Geometric Progression

Sn = [a × (r^{n − 1)}] ÷ (r − 1), where r ≠ 1

The sum of an infinite Geometric Progression

S∞ = a ÷ (1 − r), valid only when |r| < 1.
If |r| ≥ 1, the series diverges to infinity or negative infinity.

3 Basic Examples

1. Find the 10^th term of the series: 5, 10, 20…
2. Find the sum of the first 5 terms in the series: 3, 6, 12…
3. Find the sum of all terms in the series: 100, 20, 4, 4/5…

Example 1

We need to find the 10th term when a = 5 and r = 2.
t10 = 5 × 2^9

= 5 × 512

= 2560

Example 2

We need to find the sum of the first 5 terms when a = 3 and r = 2.
Sn = [3 × (2^5 − 1)] ÷ (2 − 1)

= [3 × 31] ÷ 1

= 93

Example 3

We need to find the sum of an infinite GP with a = 100 and r = 1/5.
S∞ = 100 ÷ (1 − 1/5)

= 100 × 5/4

= 125

The Larger Insight

Geometric progressions show how fast growth or decline can be managed through structured formulas. Whether the terms expand rapidly or diminish to fractions, the progression follows a predictable order. Recognizing this order and applying the formulas with confidence is what the GMAT truly tests. Developing this understanding through a GMAT preparation course helps you see clarity in what initially appears overwhelming, reinforcing the discipline of reducing complex problems into structured steps.

Parting Thought

Geometric progressions remind us that growth and decline in life often follow multiplicative patterns, not simple steps. Small choices can expand into remarkable results, just as a modest ratio transforms a sequence into something vast. GMAT preparation mirrors this truth: steady practice compounds into clarity and speed, much like terms building upon each other. The MBA applications journey, too, reflects this principle—each thoughtful essay, recommendation, and conversation amplifies your candidacy in ways that go beyond the immediate effort. In life, recognizing how repeated actions multiply over time is the key to building enduring strength and meaningful progress.