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Special Series in Sequences and Series

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Question

If T_n denotes the nth term of the series $2 + 3 + 6 + 11 + 18 + \dots$ then T₅₀ is

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Solution

Let, $S = 2 + 3 + 6 + 11 + 18 + \dots + T_{n}$
or $S = 2 + 3 + 6 + 11 + \dots + T_{n}$
On subtracting, we get
$0 = 2 + 1 + 3 + 5 + 7 + \dots - T_{n}$
$\Rightarrow T_{n} = 2 + [1 + 3 + 5 + \dots (n - 1) term]$
$\begin{array}{l} = 2 + \frac{n - 1}{2} [2 + (n - 1 - 1) 2] \\ \begin{aligned} F_{n} & = 2 + (n - 1)^{2} \\ ∴ T_{50} & = 2 + (50 - 1)^{2} = 49^{2} + 2 \end{aligned} \end{array}$

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