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Quantitative Aptitude · Sequence and Series

Geometric Progression (GP) for CA Foundation

Updated 1 October 2026 · Fact-checked

A geometric progression is a sequence where each term is the previous term multiplied by a fixed number r, the common ratio. The nth term is a·r^(n−1). The sum of n terms is a(rⁿ − 1) ÷ (r − 1) for r ≠ 1. Find a and r first, then apply the formula.

Understand Geometric Progression (GP)

A geometric progression (GP) is a sequence in which you get every term by multiplying the previous term by the same fixed number. That number is the common ratio, written r. Example: 3, 6, 12, 24 has r = 2.

To find r, divide any term by the term before it: r = T₂ ÷ T₁. The ratio must be the same for every pair of consecutive terms. If it is not, the sequence is not a GP. The ratio can be a fraction (1/2) or negative (−3). A negative ratio makes the signs alternate.

The first term is a. So the terms are a, ar, ar², ar³ and so on. The nth term is a·r^(n−1). The power of r is always one less than the term number.

AP and GP differ in how they grow. An AP adds a fixed number each step (common difference). A GP multiplies by a fixed number each step (common ratio). So an AP grows in a straight line, while a GP grows or shrinks by a constant percentage. This is why GP is used for compound interest, growth and depreciation.

Three useful properties: the square of a middle term equals the product of its neighbours (b² = ac for three terms in GP). In a finite GP, the product of terms equally far from the two ends is constant. If you multiply or divide every term of a GP by a non-zero constant, it is still a GP.

Key formulas to remember

Common ratio
r = T₂ ÷ T₁ = T₃ ÷ T₂
Must be the same for every consecutive pair. Terms are non-zero.
nth term
Tₙ = a·r^(n−1)
The power is n − 1, not n.
Sum of n terms (r > 1)
Sₙ = a(rⁿ − 1) ÷ (r − 1)
Usually used when r is greater than 1, because the denominator is then positive. Valid for any r ≠ 1.
Sum of n terms (r < 1)
Sₙ = a(1 − rⁿ) ÷ (1 − r)
Usually used when r is less than 1 (r ≠ 1), because the denominator is then positive. It gives the same result as the other form.
Sum when r = 1
Sₙ = n·a
All terms are equal, so the formulas above do not apply.
Three terms in GP
b² = ac
b is the middle term. For a,b,c all non-zero in GP. Useful for finding a missing term.
Three terms in GP (assumed form)
a/r, a, ar
Handy when the product of three terms is given: the product is a³.
nth term from the end of a finite GP
l ÷ r^(n−1)
l is the last term. Equivalent to a GP with first term l and ratio 1/r.

How to solve Geometric Progression (GP) questions

Use this method for any GP question. Most problems reduce to finding a, r and n.

  1. 1Check that the sequence is a GP by dividing consecutive terms. The ratio must be equal.
  2. 2Write down what is given and what is asked: a, r, n, Tₙ or Sₙ.
  3. 3If a and r are not given directly, form equations from the given terms using Tₙ = a·r^(n−1). Divide one equation by another to cancel a and get r.
  4. 4Solve for r first, then a, then n if needed. If r comes out as a power equation, express both sides with the same base.
  5. 5Pick a sum formula. Both (rⁿ − 1) ÷ (r − 1) and (1 − rⁿ) ÷ (1 − r) are equal for any r ≠ 1. For ease, use the first when r > 1 and the second when r < 1, so the denominator is positive.
  6. 6Substitute carefully, calculate, and check that the answer is reasonable. For r > 1 the sum must exceed the first term for positive a.
  7. 7For word problems, identify the starting amount as a and the growth or fall factor as r. A rise of 10% means r = 1.1. A fall of 10% means r = 0.9.

Quickest way: Divide, then plug into the options

When to use it: Use this in the MCQ paper when a question gives two terms or asks for a term or small sum. It saves time and limits calculation errors.

  1. Get r by dividing two terms. If the terms are k places apart, r^k equals their ratio.
  2. For small n (up to about 5), just list the terms by repeated multiplication. This is faster than using the formula and avoids power mistakes.
  3. For sums with small n, add the listed terms directly.
  4. Eliminate options quickly. For n ≥ 2, a positive a and r > 1 give a sum greater than n·a. Alternating signs mean r is negative.
  5. Check the last digit of the options against the last digit of your answer. Often only one option matches.
  6. If a sum needs a large power such as 2¹⁰ or 3⁸, skip it at first and return to it after the easier questions, because 0.25 is lost per wrong answer.

Common mistakes in Geometric Progression (GP)

  • Writing Tₙ = a·rⁿ instead of a·r^(n−1)

    Students link the term number directly to the power.

    Fix: Test with n = 1. The first term must be a, so the power must be 0. Hence n − 1.

  • Finding r by subtracting terms

    The habit from AP, where you find the common difference by subtraction.

    Fix: In a GP always divide: r = T₂ ÷ T₁.

  • Making a sign or arithmetic slip in the sum formula

    Students mix the numerator and denominator of the two forms, for example writing (rⁿ − 1) in the numerator with (1 − r) in the denominator, or drop a negative sign.

    Fix: Both (rⁿ − 1) ÷ (r − 1) and (1 − rⁿ) ÷ (1 − r) are equal for any r ≠ 1, so neither is wrong. Keep the numerator and denominator in the same form. Choosing the form with a positive denominator is only a convenience that helps you avoid sign slips.

  • Applying the sum formula when r = 1

    The formula is memorised without its condition. It gives 0 ÷ 0.

    Fix: If r = 1, all terms are equal and Sₙ = n·a.

  • Ignoring the negative root of r

    When r² = 4, students write only r = 2.

    Fix: A negative root arises only when r raised to an even power is given. For example, r² = 4 gives r = ±2. An odd power such as r³ = 8 gives only r = 2. When you get two roots, check each against the conditions in the question and keep only those that fit.

  • Treating a percentage fall as r = 0.1 or growth as r = 0.1

    Confusing the rate with the multiplier.

    Fix: Growth of 10% gives r = 1.10. A fall of 10% gives r = 0.90. The multiplier is 1 ± rate.

Worked examples

Example 1

The 3rd term of a GP is 12 and the 6th term is 96. What is the 8th term? (A) 192 (B) 384 (C) 768 (D) 512

Show the solution
  1. T₃ = a·r² = 12 and T₆ = a·r⁵ = 96.
  2. Divide: r³ = 96 ÷ 12 = 8, so r = 2.
  3. Find a: a·4 = 12, so a = 3.
  4. T₈ = a·r⁷ = 3 × 128 = 384.
  5. Check by listing from T₆: T₇ = 192, T₈ = 384.

Answer: (B) 384

Example 2

What is the sum of the first 6 terms of the GP 2, 6, 18, ...? (A) 728 (B) 364 (C) 1,458 (D) 546

Show the solution
  1. a = 2 and r = 6 ÷ 2 = 3, with n = 6.
  2. Since r > 1, use S₆ = a(r⁶ − 1) ÷ (r − 1), which has a positive denominator.
  3. 3⁶ = 729, so r⁶ − 1 = 728.
  4. S₆ = 2 × 728 ÷ 2 = 728.
  5. Check by adding: 2 + 6 + 18 + 54 + 162 + 486 = 728.

Answer: (A) 728

Example 3

A machine costing ₹1,00,000 loses value so that each year its value is 80% of the previous year's value. What is its value at the end of 3 years? (A) ₹80,000 (B) ₹64,000 (C) ₹51,200 (D) ₹40,960

Show the solution
  1. Each year the value is multiplied by 0.8, so this is a GP with r = 0.8.
  2. Value after 3 years = 1,00,000 × 0.8³.
  3. 0.8³ = 0.512.
  4. Value = 1,00,000 × 0.512 = ₹51,200.
  5. Check year by year: 80,000, then 64,000, then 51,200.

Answer: (C) ₹51,200

Exam tips

  • Questions often give two terms and ask for another. Divide the two equations to get r straight away.
  • For small n, listing terms by multiplication is usually faster and safer than powers.
  • Three-number problems are common. Take the numbers as a/r, a, ar when the product is given, and use b² = ac when you need a missing middle term.
  • Watch for AP-or-GP traps. Check whether the difference or the ratio is constant before applying any formula.
  • If the calculation needs a large power, leave the question for the end. A wrong answer costs 0.25 marks.

Practice questions from Sequence and Series

Geometric Progression (GP): frequently asked questions

What is the difference between AP and GP?

In an AP you add a fixed common difference to get the next term. In a GP you multiply by a fixed common ratio. The nth term of an AP is a + (n − 1)d and of a GP is a·r^(n−1).

How do I find the common ratio of a GP?

Divide any term by the term just before it. For example, in 5, 15, 45 the ratio is 15 ÷ 5 = 3. Check one more pair to confirm it is the same.

Which sum formula should I use for a GP?

Both Sₙ = a(rⁿ − 1) ÷ (r − 1) and Sₙ = a(1 − rⁿ) ÷ (1 − r) give the same value for any r ≠ 1. For convenience, use the first when r > 1 and the second when r < 1, so the denominator is positive. If r = 1, the sum is simply n × a.

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

Yes. A fraction between 0 and 1 makes the terms shrink. A negative ratio makes the signs alternate, as in 2, −6, 18, −54. The ratio cannot be zero for a GP.