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Question

If each of $$n$$ numbers $$x_i = i$$ is replaced by $$(i+1)x_i$$, then the new mean is


A
(n+1)(n+2)n
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B
n+1
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C
(n+1)(n+2)3
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D
none of these
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Solution

The correct option is C $$\displaystyle\frac{(n+1)(n+2)}{3}$$
Required mean $$=\displaystyle \frac{\sum (1+i)x_i}{n}= \frac{\sum (1+i)i}{n} =\frac{\sum (i^2+i)}{n}=\frac{\sum i^2+\sum i}{n}$$
$$\displaystyle \quad =\frac{\frac{1}{6}n(n+1)(2n+1)+\frac{1}{2}n(n+1)}{n}=\frac{(n+1)(n+2)}{3}$$

Mathematics

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