Sum of AP and sum of first n natural numbers

CAT 2019 Slot 2 · Quantitative Ability · Medium · Sequence & series

This is a medium Quantitative Ability question from the CAT 2019 Slot 2 paper. It tests Sequence & series. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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If $(2n+1) + (2n+3) + (2n+5) + \dots + (2n+47) = 5280$ , then what is the value of $1+2+3+\dots+n$ ?

[TITA]

Answer

4851

Explanation

The terms are of the form $2n + (2k-1)$ for $k = 1, 2, \dots, 24$ (since $2k - 1 = 47 \implies k = 24$ ).

Sum $= 24 \times 2n + (1 + 3 + 5 + \dots + 47) = 48n + 24^2 = 48n + 576$ . $48n + 576 = 5280$ $48n = 4704 \implies n = 98$

We need $1 + 2 + \dots + 98 = \frac{98 \times 99}{2} = 49 \times 99 = 4851$ .

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