Find the sum of series $1 + (2) (3) + (4) + (5) (6) + 7 + . . . .$ upto $50$ term.

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Solution

Sum of $1 + (2) (3) + 4 + (5) (6) + 7 - - - - - -$ up to 50 terms.
$1 + 4 + 7 + 10 + 13 - - - - - - - - - + 73$
$+ (2) (3) + (5) (6) + (8) (9) - - - - -$
$\Rightarrow 1 + 4 + 7 + 10 + 13 - - - - + 73$
$= 73 = 1 + (n - 1) 3$
$= 73 = 1 + 3 n - 3$
$= 72 + 3 = 3 n$
$n = 25$
$= \frac{25}{2} (2 + (24) 3)$
$= \frac{25}{2} (2 + 72)$
$= 25 \times 37$
$= 925$

$\Rightarrow (2) (3) + (5) (6) + (8) (9) - - - - + (74) (75)$
$\Rightarrow n = 1 \sum 25 (3^{n} - 1) (3^{n})$
$\Rightarrow 3^{2 n} - 3^{n}$
$n = 1 \sum 25 9^{n} - 3 n$
$n = 1 \sum 25 9 n - n = 1 \sum 25 3^{n}$
$\Rightarrow \frac{9^{26} + 4 (3^{26}) - 12}{8} + 925$

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