Find the sum of the series 1 + 3 + 7 + 15 + 3 | vedclass

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(B) Let the $n^{th}$ term be $T_n$ and the sum of $n$ terms be $S_n$.$S_n = 1 + 3 + 7 + 15 + \dots + T_n \quad (i)$$S_n = 1 + 3 + 7 + \dots + T_{n-1}

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$2^{n+1} - n - 2$

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(B) Let the $n^{th}$ term be $T_n$ and the sum of $n$ terms be $S_n$.$S_n = 1 + 3 + 7 + 15 + \dots + T_n \quad (i)$$S_n = 1 + 3 + 7 + \dots + T_{n-1} + T_n \quad (ii)$Subtracting $(ii)$ from $(i)$:$0 = 1 + [2 + 4 + 8 + 16 + \dots + (T_n - T_{n-1})] - T_n$$T_n = 1 + (2 + 4 + 8 + \dots + 2^{n-1})$$T_n = 1 + \frac{2(2^{n-1} - 1)}{2 - 1} = 1 + 2^n - 2 = 2^n - 1$Now,$S_n = \sum_{k=1}^n T_k = \sum_{k=1}^n (2^k - 1) = \sum_{k=1}^n 2^k - \sum_{k=1}^n 1$$S_n = \frac{2(2^n - 1)}{2 - 1} - n = 2^{n+1} - 2 - n$

Find the sum of the series $1 + 3 + 7 + 15 + 31 + \dots$ up to $n$ terms.

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