The arithmetic mean of the sequence 1, 2, 4, | vedclass

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Question

(B) The given sequence is $1, 2, 4, 8, 16, \ldots, 2^n$.This is a geometric progression $(G.P.)$ with the first term $a = 1$ and common ratio $r = 2$.

Vedclass · Accepted Answer

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

Vedclass · Answer

(B) The given sequence is $1, 2, 4, 8, 16, \ldots, 2^n$.This is a geometric progression $(G.P.)$ with the first term $a = 1$ and common ratio $r = 2$.The number of terms in the sequence is $n+1$ (since the powers of $2$ range from $2^0$ to $2^n$).The sum of the terms of the $G.P.$ is given by $S = \frac{a(r^{n+1} - 1)}{r - 1} = \frac{1(2^{n+1} - 1)}{2 - 1} = 2^{n+1} - 1$.The arithmetic mean $(A.M.)$ is defined as the sum of the terms divided by the number of terms.Therefore,$A.M. = \frac{S}{n+1} = \frac{2^{n+1} - 1}{n+1}$.

The arithmetic mean of the sequence $1, 2, 4, 8, 16, \ldots, 2^n$ is:

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