$99^{t h}$ term of the series $2 + 7 + 14 + 23 + 34 + . . . .$ is

9998

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9999

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10000

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None of these

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Solution

The correct option is A $9998$
Let $S = 2 + 7 + 14 + 23 + 34 + . . . . . . . . . . . . . . . . . . . . . . . . . . . + T_{n - 1} + T_{n}$
$S = 2 + 7 + 14 + 23 + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + T_{n - 1} + T_{n}$
$0 = 2 + [5 + 7 + 9 + 11 + . . . . . . . . . . . . . . . . . . . . . + (n - 1) t e r m s] - T_{n}$
$T_{n} = 2 + \frac{(n - 1)}{2} [2 \times 5 + (n - 1 - 1) \times 2] = 2 + (n - 1) [5 + (n - 2)] T_{99} = 2 + (99 - 1) [5 + (99 - 2)] = 2 + 98 (5 + 97) = 2 + 98 \times 102 = 9998$
$∴$ $99^{t h}$ term of the series is=9998

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