---
title: "Induction: sum of geometric progression"
subject: "Discrete Structure"
chapter: 1
key: "semester_4/discrete_structure:induction-geometric-series-sum"
source: https://www.bctengineer.com/q/semester_4/discrete_structure/induction-geometric-series-sum
site: BCT Engineering
---

# Induction: sum of geometric progression

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2074 Bhadra, 4 marks
- Video: [Proof of Sum of Geometric Series Formula (using proof by induction)](https://www.youtube.com/watch?v=NFIymrfOkDU) — Tulla Maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/induction-geometric-series-sum
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Use mathematical induction to prove the formula for the sum of a finite number of terms of Geometric Progression: Σ(j=0 to n) a·r^j = a + ar + ar² + .... + ar^k = (a·r^(n+1) − a)/(r − 1), when r ≠ 1, where n is non-negative integer.

## Short answer

Basis n=0: a = (ar−a)/(r−1). Assume sum to k is (ar^(k+1)−a)/(r−1). Adding ar^(k+1) gives (ar^(k+2)−a)/(r−1). By induction, true for all n≥0.

## Answer

(The last term written as ar^k in the question should be arⁿ, since the sum runs from j = 0 to n.)

**To prove.** P(n): a + ar + ar² + … + arⁿ = (arⁿ⁺¹ − a)/(r − 1), for r ≠ 1 and all integers n ≥ 0.

#### Basis step (n = 0)
- LHS = ar⁰ = a.
- RHS = (ar − a)/(r − 1) = a(r − 1)/(r − 1) = a, since r ≠ 1.
- P(0) is true.

#### Inductive hypothesis
Assume P(k) is true: a + ar + … + arᵏ = (arᵏ⁺¹ − a)/(r − 1).

#### Inductive step
Show P(k+1): a + ar + … + arᵏ + arᵏ⁺¹ = (arᵏ⁺² − a)/(r − 1).

```text
LHS = (arᵏ⁺¹ − a)/(r − 1) + arᵏ⁺¹            (hypothesis)
    = [arᵏ⁺¹ − a + arᵏ⁺¹ (r − 1)]/(r − 1)
    = [arᵏ⁺¹ − a + arᵏ⁺² − arᵏ⁺¹]/(r − 1)
    = (arᵏ⁺² − a)/(r − 1) = RHS
```

#### Conclusion
By mathematical induction, the formula holds for every non-negative integer n. ∎

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. At n = 0 the RHS (ar − a)/(r − 1) equals?
   A. ar
   B. a
   C. 0
   D. a(r+1)
   Answer: B. a(r − 1)/(r − 1) = a since r ≠ 1. (Key point: Basis)

2. The inductive step adds which term to the hypothesis sum?
   A. arᵏ
   B. arᵏ⁺¹
   C. arᵏ⁺²
   D. a
   Answer: B. P(k+1) has one more term, arᵏ⁺¹. (Key point: Term added)

3. arᵏ⁺¹ − a + arᵏ⁺¹(r − 1) simplifies to?
   A. arᵏ⁺¹ − a
   B. arᵏ⁺² − a
   C. 2arᵏ⁺¹ − a
   D. arᵏ⁺² − arᵏ⁺¹
   Answer: B. arᵏ⁺¹(r−1) = arᵏ⁺² − arᵏ⁺¹, cancelling the arᵏ⁺¹ term. (Key point: Algebra)

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