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Q. The sum of the series $1+2 \times 3+4+5 \times 6+7+8 \times 9+$ ... up to $50$ terms is

Sequences and Series

Solution:

$S_{50}= \displaystyle \sum_{r=1}^{25}[(3 r-2)+(3 r-1)(3 r)]$
$= \displaystyle \sum_{r=1}^{25}\left(9 r^{2}-2\right)$
$=\frac{9}{6}(25)(26)(51)-50=49675$