The sum $1 + 3 + 7 + 15 + 31 + . .$ .to n terms is

2^{n} - 2 - n

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2^{n - 1} - 2 - n

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2^{n + 1} - 2 - n

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None of these

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Solution

The correct option is B $2^{n - 1} - 2 - n$
Given that,
$1 + 3 + 15 + 31 + . . . . .$ up to $n$ 4 termsup to $n$ terms

$S u m = 1 + 2^{2} - 1 + 2^{3} - 1 + 2^{4} - 1 + . . . . . . . . .$ up to $n$ terms
$S u m = 1 - 1 - 1 - 1 - . . . . . . u p t o (n - 1) t e r m + 2^{2} + 2^{3} + 2^{4} . . . . . .$ up to $n - 1$ terms
$S u m = 1 - (n - 1) + 2^{2} \frac{2^{n - 1} - 1}{2 - 1}$

$S u m = 1 - n + 1 + 2^{2} 2^{n - 1} - 2^{2}$
$S u m = 2 - n + 2^{n + 1} - 4$

$S u m = 2^{n + 1} - n - 2$

Hence, this is the answer.

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