First slide

Special Series in Sequences and Series

Question

$The sum of the first n terms of the series 1^{2} + {2.2}^{2} + 3^{2} + {2.4}^{2} + 5^{2} + .$ $Is \frac{n (n + 1)^{2}}{2}, when n is even, when 'n' is odd sum is$

Moderate

A

${[\frac{n(n+1)}{2}]}^{2}$
B

$\frac{n^{2} (n+1)}{2}$
C

$\frac{{n(n+1)}^{2}}{4}$
D

$\frac{3n(n+1)}{2}$

Solution

If n is odd, the required sum is
$\begin{array}{l} 1^{2} + {2.2}^{2} + 3^{2} + 2 \cdot 4^{2} + 5^{2} + . . + 2 (n - 1)^{2} + n^{2} = \frac{(n - 1) n^{2}}{2} + n^{2} \\ = \frac{n^{2} (n + 1)}{2} \end{array}$

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