Find the sum of n terms of the series 1 + 3 + | vedclass

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(B) Let the $n$-th term be $t_n$. The series is $1, 3, 7, 15, 31, \dots$The terms can be written as $(2^1 - 1), (2^2 - 1), (2^3 - 1), (2^4 - 1), (2^5

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

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(B) Let the $n$-th term be $t_n$. The series is $1, 3, 7, 15, 31, \dots$The terms can be written as $(2^1 - 1), (2^2 - 1), (2^3 - 1), (2^4 - 1), (2^5 - 1), \dots$Thus,the $n$-th term is $t_n = 2^n - 1$.The sum of $n$ terms $S_n = \sum_{k=1}^{n} t_k = \sum_{k=1}^{n} (2^k - 1)$.$S_n = \sum_{k=1}^{n} 2^k - \sum_{k=1}^{n} 1$.The first part is a geometric progression with first term $a = 2$ and common ratio $r = 2$.Sum of $n$ terms of $GP$ $= \frac{a(r^n - 1)}{r - 1} = \frac{2(2^n - 1)}{2 - 1} = 2(2^n - 1) = 2^{n+1} - 2$.Therefore,$S_n = (2^{n+1} - 2) - n = 2^{n+1} - n - 2$.

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

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