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

# Induction: alternating sum of squares

Discrete Structure · Chapter 1: Logic, Induction and Reasoning

- Asked in: 2075 Bhadra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/discrete_structure/induction-alternating-squares
- Whole chapter: https://www.bctengineer.com/md/semester_4/discrete_structure/chapter-1

## Question

Use mathematical induction to verify: 1² − 2² + 3² − 4² + ... + (−1)^(n+1) n² = (−1)^(n+1) n(n+1)/2

## Short answer

Basis n=1: 1² = 1 = (−1)²·1·2/2. Assume sum to k equals (−1)^(k+1)k(k+1)/2. Adding (−1)^(k+2)(k+1)² and factoring yields (−1)^(k+2)(k+1)(k+2)/2. By induction, the formula holds.

## Answer

**To prove.** P(n): 1² − 2² + 3² − 4² + … + (−1)ⁿ⁺¹ n² = (−1)ⁿ⁺¹ n(n+1)/2 for all positive integers n.

#### Basis step (n = 1)
- LHS = 1² = 1.
- RHS = (−1)² · 1 · 2 / 2 = 1.
- So P(1) is true.

#### Inductive hypothesis
Assume P(k) is true for some k ≥ 1:

1² − 2² + … + (−1)ᵏ⁺¹ k² = (−1)ᵏ⁺¹ k(k+1)/2

#### Inductive step
We show P(k+1): the sum up to (−1)ᵏ⁺² (k+1)² equals (−1)ᵏ⁺² (k+1)(k+2)/2.

```text
LHS = [1² − 2² + … + (−1)ᵏ⁺¹ k²] + (−1)ᵏ⁺² (k+1)²
    = (−1)ᵏ⁺¹ k(k+1)/2 + (−1)ᵏ⁺² (k+1)²     (hypothesis)
```
Since (−1)ᵏ⁺¹ = −(−1)ᵏ⁺², take (−1)ᵏ⁺² (k+1) as common factor:
```text
    = (−1)ᵏ⁺² (k+1) [ −k/2 + (k+1) ]
    = (−1)ᵏ⁺² (k+1) (k+2)/2
    = RHS of P(k+1)
```

#### Conclusion
P(1) is true and P(k) → P(k+1) for all k ≥ 1, so by mathematical induction P(n) holds for all positive integers n. ∎

## Check yourself

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

1. What is the basis step for this formula?
   A. n = 0
   B. n = 1: LHS = 1, RHS = (−1)²·1·2/2 = 1
   C. n = 2
   D. n = k+1
   Answer: B. For n = 1 both sides equal 1. (Key point: Basis)

2. What term is added to go from k to k+1?
   A. (−1)ᵏ⁺¹ (k+1)²
   B. (−1)ᵏ⁺² (k+1)²
   C. (−1)ᵏ (k+1)
   D. (k+1)²
   Answer: B. The (k+1)th term is (−1)ᵏ⁺² (k+1)². (Key point: New term)

3. What identity lets us factor out (−1)ᵏ⁺² (k+1)?
   A. (−1)ᵏ⁺¹ = (−1)ᵏ⁺²
   B. (−1)ᵏ⁺¹ = −(−1)ᵏ⁺²
   C. (−1)ᵏ⁺² = 2(−1)ᵏ⁺¹
   D. (−1)ᵏ = 1
   Answer: B. Consecutive powers of −1 have opposite signs. (Key point: Sign trick)

4. The bracket −k/2 + (k+1) simplifies to?
   A. (k+1)/2
   B. (k+2)/2
   C. (2k+1)/2
   D. k/2
   Answer: B. −k/2 + k + 1 = k/2 + 1 = (k+2)/2. (Key point: Algebra)

## Related in Discrete Structure

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