Which of the following is a value of n if (_k=1^n(-1)^k-1k)^2-_k=1^n(-1)^k-1k^2+2450=0?

B. 99. Step 1 (odd n=2m-1): 1-2+3-…+(2m-1)=m Step 2: 1^2-2^2+3^2-…+(2m-1)^2: pair (2j-1)^2-(2j)^2=-(4j-1) for j=1 to m-1, then add (2m-1)^2. Result =m(2m-1). Step 3: Equation: m^2-m(2m-1)+2450=0 -m^2+m+2450=0 m^2-m-2450=0 Step 4: (m-50)(m+49)=0 m=50, so n=2m-1=99. Check even n=2m: first sum =-m, second =-m(2m+1); equation m^2+m(2m+1)+2450=0 has no positive solution.

Question 4 of 42 · Arithmetic & Geometric Progression

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NIMCET 2026

Which of the following is a value of $n$ if $(\sum_{k = 1}^{n} (- 1)^{k - 1} k)^{2} - \sum_{k = 1}^{n} (- 1)^{k - 1} k^{2} + 2450 = 0$ ?

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Answer: Option B — 99

Detailed Solution

Step 1 (odd $n = 2 m - 1$ ): $1 - 2 + 3 - \dots + (2 m - 1) = m$

$1^{2} - 2^{2} + 3^{2} - \dots + (2 m - 1)^{2}$ : pair $(2 j - 1)^{2} - (2 j)^{2} = - (4 j - 1)$ for $j = 1$ to $m - 1$ , then add $(2 m - 1)^{2}$ . Result $= m (2 m - 1)$ .

Equation: $m^{2} - m (2 m - 1) + 2450 = 0 \Rightarrow - m^{2} + m + 2450 = 0 \Rightarrow m^{2} - m - 2450 = 0$

$(m - 50) (m + 49) = 0 \Rightarrow m = 50$ , so $n = 2 m - 1 = 99$ .

Check even $n = 2 m$ : first sum $= - m$ , second $= - m (2 m + 1)$ ; equation $m^{2} + m (2 m + 1) + 2450 = 0$ has no positive solution.

Quick tip

For odd $n$ : $\sum (- 1)^{k - 1} k = \frac{n + 1}{2}$ , $\sum (- 1)^{k - 1} k^{2} = \frac{n ( n + 1 )}{2}$

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Arithmetic & Geometric Progression — Preparation

This question tests your understanding of Arithmetic & Geometric Progression, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

An arithmetic progression (AP) is a sequence where each term differs from the previous one by a constant amount, the common difference. A geometric progression (GP) is a sequence where each term is the previous one multiplied by a constant, the common ratio. This topic covers the nth term and sum of n terms of an AP and GP, the sum of an infinite GP, arithmetic and geometric means, and word problems. Key formulas are the AP nth term a + (n - 1)d, the AP sum n/2 [2a + (n - 1)d], the GP nth term ar^(n-1), and the infinite GP sum a/(1 - r) for |r| < 1. A useful trick is to choose symmetric terms, such as a - d, a, a + d for three terms of an AP, which simplifies many questions. Practise the AP and GP questions below, each with a step-by-step solution.

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Arithmetic & Geometric Progression — Preparation

Which of the following is a value of $n$ if $(\sum_{k = 1}^{n} (- 1)^{k - 1} k)^{2} - \sum_{k = 1}^{n} (- 1)^{k - 1} k^{2} + 2450 = 0$ ?