Question

the Arithmetic mean of the series 1, 2, 4, 8, 16.......2^n​

Answer 1

Step-by-step explanation:

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Answer 2

The arithmetic mean is $\frac{2^{n+1}-1}{n+1}$

Step-by-step explanation:

Given : The series  $1, 2, 4, 8, 16.......2^n$.

To find : The arithmetic mean of the series ?

Solution :

Arithmetic mean is defined as sum of the terms of series divided by total number of terms.

$AM=\frac{\sum x_n}{n}$

i.e. $AM=\frac{1+2+4+8+...+2^n}{n+1}$

$AM=\frac{2^0+2^1+2^2+2^3+...+2^n}{n+1}$

The numerator is forming a geometric progression with first term a=1 and common ratio r=2.

Applying GP sum formula, $S_n=\frac{a(r^n-1)}{r-1}$

$AM=\frac{\frac{1(2^n-1)}{2-1}}{n+1}$

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

Therefore, the arithmetic mean is $\frac{2^{n+1}-1}{n+1}$

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Arithmetic mean of series a+(a+d)+(a+d)+(a+2d)......(a+2nd)is

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