Fundamentals of Business Mathematics and Statistics · Arithmetic Progression and Geometric Progression

Geometric Progression: nth Term and Common Ratio Explained

Updated 10 October 2026 · Fact-checked

A geometric progression (GP) is a sequence where each term is the previous term multiplied by a fixed number, the common ratio r = T₂ ÷ T₁. The nth term is Tₙ = a·rⁿ⁻¹. To solve a question, find a and r, then substitute. For three numbers in GP, take a/r, a, ar.

Understand Geometric Progression: nth Term and Common Ratio

A geometric progression (GP) is a list of numbers in which you get every term by multiplying the one before it by the same fixed number. That fixed number is the common ratio, written r.

Take 3, 6, 12, 24. Each term is twice the previous one, so r = 2. In 81, 27, 9, 3, each term is one-third of the previous one, so r = 1/3. If r is negative, the signs alternate, as in 2, -4, 8, -16, where r = -2.

To find r, divide any term by the term just before it: r = T₂ ÷ T₁ = T₃ ÷ T₂. If these ratios are not equal, the sequence is not a GP. This is the test the exam uses.

The first term is a. The second is ar, the third is ar², the fourth is ar³. The power of r is always one less than the term number. So the nth term is Tₙ = a·rⁿ⁻¹.

If r > 1 the terms grow (for positive a). If 0 < r < 1 they shrink. Business uses are growth of sales by a fixed percentage each year, or a machine losing a fixed percentage of value each year.

Key formulas to remember

Common ratio: r = T₂ ÷ T₁ = T₃ ÷ T₂ = Tₙ ÷ Tₙ₋₁; Divide a term by the one before it. All such ratios must be equal for a GP. The first term must not be zero.
nth term of a GP: Tₙ = a · rⁿ⁻¹; a is the first term, r the common ratio. The power is n − 1, not n.
Term from the end of a finite GP: nth term from the end = l · (1/r)ⁿ⁻¹; l is the last term. Treat the GP in reverse with ratio 1/r.
Relation between two terms: Tₘ ÷ Tₙ = r^(m − n); Useful when two terms are given and a is not needed.
Three numbers in GP: a/r, a, ar; Product = a³. Choose this form when the product is given.
Condition for three numbers in GP: b² = ac; If a, b, c are in GP, then b is the middle term and b² = ac.

How to solve Geometric Progression: nth Term and Common Ratio questions

Use this method for any question on the nth term or common ratio of a GP.

1Check that the sequence is a GP by confirming T₂ ÷ T₁ = T₃ ÷ T₂.
2Write down what is given: a, r, n, or particular terms such as T₃ and T₆.
3Find r. If two terms are given, divide them: Tₘ ÷ Tₙ = r^(m − n), then take the root.
4If r is found from an even power, remember it can be positive or negative. Check conditions in the question, such as all terms positive.
5Find a by putting r back into one given term, if a is needed.
6Substitute into Tₙ = a·rⁿ⁻¹ and calculate.
7For selecting terms, use a/r, a, ar for three terms, and a/r³, a/r, ar, ar³ for four terms. Use the sum and product to get a and r.
8Verify your answer by checking the ratio or the given sum or product.

Quickest way: Divide two terms to get r, skip a

When to use it: When two terms of a GP are given and you need a third term. This is common in MCQs.

Do not find a. Use Tₘ ÷ Tₙ = r^(m − n).
Get r from the two given terms.
Move from the nearest given term: Tₖ = Tₙ · r^(k − n).
For three numbers in GP with product given, put the middle term = cube root of the product immediately.
For options, check the answer by testing the ratio of consecutive terms.

Common mistakes in Geometric Progression: nth Term and Common Ratio

Using Tₙ = a·rⁿ instead of a·rⁿ⁻¹.
Students remember 'a times r power n' and forget the first term has no r.
Fix: Check with n = 1: the answer must be a. Only rⁿ⁻¹ gives that.
Finding r by subtracting terms.
It is mixed up with common difference in an AP.
Fix: For a GP always divide. For AP always subtract.
Dividing a term by the next term instead of the previous term.
Students write the ratio upside down.
Fix: r = later term ÷ earlier term. For 16, 8, 4, r = 8/16 = 1/2, not 2.
Ignoring the negative root of r.
If r² = 4, students write only r = 2.
Fix: Write r = ±2 and test both against the question. If none rules one out, check whether the options contain both answers.
Choosing wrong form for three numbers when product is given.
Students use a, ar, ar² and get a hard equation.
Fix: Use a/r, a, ar so the product is a³ and r cancels.
Calculating the power incorrectly, like (1/2)⁴ as 1/8.
Rushing arithmetic on a one-hour paper.
Fix: Write the power as a product of repeated factors. (1/2)⁴ = 1/16.

Worked examples

Example 1

The 2nd term of a GP is 6 and the 5th term is 162. Find the common ratio and the 7th term.

Show the solution

T₅ ÷ T₂ = r^(5 − 2) = r³.
r³ = 162 ÷ 6 = 27, so r = 3.
T₇ = T₅ · r^(7 − 5) = 162 × 3².
T₇ = 162 × 9 = 1,458.
Check: a = T₂ ÷ r = 2. Then T₇ = 2 × 3⁶ = 2 × 729 = 1,458.

Answer: Common ratio = 3 and 7th term = 1,458.

Example 2

Three numbers are in GP. Their sum is 21 and their product is 216. Find the numbers, taking the common ratio as greater than 1.

Show the solution

Let the numbers be a/r, a, ar.
Product = a³ = 216, so a = 6.
Sum: 6/r + 6 + 6r = 21, so 6/r + 6r = 15.
Multiply by r: 6 + 6r² = 15r, so 6r² − 15r + 6 = 0.
Divide by 3: 2r² − 5r + 2 = 0, so (2r − 1)(r − 2) = 0.
r = 2 or r = 1/2. Since r > 1, r = 2.
Numbers: 6/2 = 3, 6, 6 × 2 = 12.
Check: 3 + 6 + 12 = 21 and 3 × 6 × 12 = 216.

Answer: The numbers are 3, 6 and 12. If r = 1/2 you get the same numbers in reverse order.

Exam tips

Always test the ratio T₂ ÷ T₁ and T₃ ÷ T₂ first. Questions often hide a sequence that is not a GP among the options.
When the product of three numbers in GP is given, the middle number is the cube root of the product. This can often give the answer without any further work.
In MCQs, plug the options back into the sum or product condition. This is faster than solving the quadratic.
Read whether the question says r is positive, greater than 1, or all terms are positive. It decides which root of r to keep.
Memorise small powers: 2⁷ = 128, 2¹⁰ = 1,024, 3⁵ = 243, 5⁴ = 625. They save time in nth term calculations.

Practice questions from Arithmetic Progression and Geometric Progression

Geometric Progression: nth Term and Common Ratio in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

CA FoundationGeometric Progression (GP)

Geometric Progression: nth Term and Common Ratio: frequently asked questions

What is the nth term formula of a GP?

Tₙ = a·rⁿ⁻¹, where a is the first term, r is the common ratio and n is the term number. For example, with a = 5 and r = 2, the 4th term is 5 × 2³ = 40.

How do you find the common ratio of a GP?

Divide any term by the term just before it, for example T₂ ÷ T₁. If two terms are not next to each other, divide them and take a root: Tₘ ÷ Tₙ = r^(m − n).

How do you find three numbers in GP with given sum and product?

Take the numbers as a/r, a, ar. The product gives a³, so you find a at once. Then put a into the sum equation to get a quadratic in r and solve it.

Can the common ratio of a GP be negative or a fraction?

Yes. A negative r makes the terms alternate in sign, and a fraction between 0 and 1 makes terms shrink. The ratio cannot be zero, and the first term should not be zero.