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

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

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n + 1

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\frac{(n + 1) (n + 2)}{3}

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none of these

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Solution

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

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

The correct option is C (n+1)(n+2)3Required mean =∑(1+i)xin=∑(1+i)in=∑(i2+i)n=∑i2+∑in=16n(n+1)(2n+1)+12n(n+1)n=(n+1)(n+2)3

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

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