99^ th term of the series 2 + 7 + 14 + 23 + 34 + ..... is - Vidyadip

Question

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

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Answer

a
(a) Let $S = 2 + 7 + 14 + 23 + 34 + ..... + {T_n}$…..(i)

and $S = \,\,\,\,\,\,\,\,\,\,2 + 7 + 14 + ................. + {T_{n - 1}} + {T_n}$ …..(ii)

From (i) and (ii), we get

$0 = 2 + [5 + 7 + 9 + 11........ + {T_n} - {T_{n - 1}}] - {T_n}$

$ \Rightarrow $ ${T_n} = 2 + \left[ {\frac{{n - 1}}{2}\{ 2 \times 5 + (n - 2)\,2\} } \right]$

$ \Rightarrow $${T_n} = 2 + (n - 1)(n + 3)$

Now put $n = 99$

$ \Rightarrow $${T_{99}} = 2 + 98 \times 102 = 9998$.

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