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  4. 99th term of the series 2 + 7 + 14 + 23 + 34 + ...........is

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Q. 99th term of the series $2 + 7 + 14 + 23 + 34 + ..........$.is

Sequences and Series

Solution:

Let $S_{n} = 2+7+14+23+34+..... +T_{n-1} +T_{n}$
Also $S_{n} = 2+7+14+23+.....+T_{n-1} +T_{n} $
$\therefore 0 = 2+ \left[5+7+9+11+......\left(n-1\right) terms\right] - T_{n} $
$ \therefore T_{n} = 2+\left[5+7+9+11+.....\left(n-1\right) terms\right]$
$ = 2+\frac{n-1}{2} \left[ 2\times5+\left(n-2\right)2\right] $
$= 2+\frac{n-1}{2} \left[10+2n-4\right] $
$ 2+\left(n-1\right)\left(n+3\right)$
$ \therefore T_{99} = 2+\left(98\right)\left(102\right)$
$ = 2+9996 = 9998$