Geometric Progression Explained — Sequence and Series (JAMB / WAEC Maths)
A geometric progression (GP) is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed number called the common ratio.
This topic is a regular feature in JAMB and WAEC Mathematics, usually tested through finding a specific term, the sum of a number of terms, or the sum to infinity. This lesson walks through all three with worked examples.
Quick takeaways
- A geometric progression (GP) is a sequence where each term is obtained by multiplying the previous term by a constant value called the common ratio (r).
- The nth term of a GP is given by the formula: Tₙ = arⁿ⁻¹, where “a” is the first term.
- The sum of the first n terms is given by: Sₙ = a(rⁿ − 1)/(r − 1), for r ≠ 1.
- The sum to infinity, S∞ = a/(1 − r), only exists when the common ratio’s absolute value is less than 1 (|r| < 1).
- Exam questions often give you two terms of a GP and ask you to find the first term and common ratio before solving the actual question — practice this two-step setup.
What is a geometric progression?
A geometric progression (GP) is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio (r).
For example, in the sequence 2, 6, 18, 54…, each term is obtained by multiplying the previous one by 3, so the common ratio is 3. This is different from an arithmetic progression, where each term is found by adding a fixed amount rather than multiplying.
How do you find the nth term of a geometric progression?
The nth term of a GP is calculated using the formula:
Tₙ = arⁿ⁻¹
Where:
- a = the first term of the sequence
- r = the common ratio
- n = the term number you want to find
For example, to find the 5th term of the sequence 2, 6, 18…, where a = 2 and r = 3: T₅ = 2(3)⁴ = 2 × 81 = 162.
How do you find the sum of the first n terms of a geometric progression?
The sum of the first n terms of a GP is calculated using:
Sₙ = a(rⁿ − 1)/(r − 1) (for r ≠ 1)
This formula lets you add up a specific number of terms without listing and adding each one individually, which becomes especially useful as n grows large.
What is the sum to infinity, and when does it exist?
The sum to infinity of a GP, written S∞, gives the total sum of an infinite number of terms, but only when this sum converges to a finite value. This only happens when the absolute value of the common ratio is less than 1 (|r| < 1), meaning each successive term gets progressively smaller, approaching zero. The formula is:
S∞ = a/(1 − r)
If |r| ≥ 1, the terms don’t shrink toward zero, and the sum to infinity doesn’t exist (it grows without bound), so always check this condition before applying the formula.
How do you solve GP questions when only two terms are given?
Many JAMB/WAEC questions give you two specific terms of a GP (not necessarily the first two) and ask you to find the first term, common ratio, or a different specific term. The general approach:
- Write out the formula for each given term using Tₙ = arⁿ⁻¹
- Divide one equation by the other to eliminate “a” and solve for “r”
- Substitute the value of “r” back into either equation to find “a”
- Use these values to answer the actual question asked
Common mistakes students make with geometric progression
- Confusing the formulas for a GP with those for an arithmetic progression (multiplication vs addition-based sequences)
- Forgetting to check that |r| < 1 before applying the sum to infinity formula
- Making sign errors when the common ratio is negative, since powers of a negative number alternate in sign
- Mixing up which term number corresponds to which exponent in the nth term formula (remember it’s rⁿ⁻¹, not rⁿ)
Frequently asked questions
Q: What is the formula for the nth term of a geometric progression? The nth term is given by Tₙ = arⁿ⁻¹, where “a” is the first term, “r” is the common ratio, and “n” is the term number.
Q: When does the sum to infinity of a geometric progression exist? The sum to infinity only exists when the absolute value of the common ratio is less than 1 (|r| < 1), since this ensures the terms shrink progressively toward zero.
Q: What is the difference between a geometric progression and an arithmetic progression? In a geometric progression, each term is found by multiplying the previous term by a constant common ratio, while in an arithmetic progression, each term is found by adding a constant common difference.
Q: How do you find the common ratio if you’re only given two non-consecutive terms? Write the formula for each given term using Tₙ = arⁿ⁻¹, then divide one equation by the other to eliminate “a” and solve directly for the common ratio “r”.
Want to see how repeated multiplication connects to exponential growth? Read our Exponential Functions lesson →
This is one topic from our full JAMB & WAEC Maths course — see everything covered →
