The arithmetic mean of the series $1, 2, 2^{2}, . . . {.2}^{n - 1}$ is

\frac{2^{n}}{n}

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\frac{(2^{n} - 1)}{n}

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\frac{(2^{n + 1})}{n}

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None of these

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Solution

The correct option is B $\frac{(2^{n} - 1)}{n}$
Sum of the given series is $= 1 + 2 + 2^{2} + . . . . . . . . . . + 2^{n - 1} = \frac{2^{n} - 1}{2 - 1} = 2^{n} - 1$
And number of terms $= n$
Hence arithmetic mean $= \frac{Sum of all the terms}{Number of terms} = \frac{2^{n} - 1}{n}$

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