JEE Main 2026 Apr 6 Shift 1, Mathematics Q5: Summation of Series

JEE Main2026Apr 6, Shift 1Mathematics

The value of $1^{3} - 2^{3} + 3^{3} - \dots + 1 5^{3}$ is:

Solution

Group the first $14$ terms into pairs of the form $(2 k - 1)^{3} - (2 k)^{3}$ for $k = 1, 2, \dots, 7$ , and add the lone last term $1 5^{3}$ .

Using $n^{3} - (n + 1)^{3} = - 3 n^{2} - 3 n - 1$ with $n = 1, 3, 5, \dots, 13$ :

$\sum_{n \in {1, 3, 5, 7, 9, 11, 13}} (- 3 n^{2} - 3 n - 1) = - 3 (1 + 9 + 25 + 49 + 81 + 121 + 169) - 3 (1 + 3 + 5 + 7 + 9 + 11 + 13) - 7$

$= - 3 (455) - 3 (49) - 7 = - 1365 - 147 - 7 = - 1519$ .

Finally add $1 5^{3} = 3375$ : total $= 3375 - 1519 = 1856$ .

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