Q16 / 77 · NTA Abhyas (JEE Main) 2020 · Mathematics · Statistics

The mean of $n$ observations is $\bar{X} .$ If the first observation is increased by $1^{2},$ second by $2^{2}$ and so on, then the new mean is

Solution

Let,

x_{1}, x_{2}, . . . . . x_{n}

n

observations.
Then,

\bar{X} = \frac{1}{n} Σ x_{i} .

Let,

y_{1} = x_{1} + 1^{2}, y_{2} = x_{2} + 2^{2},

y_{3} = x_{3} + 3^{2}, . . . ., y_{n} = x_{n} + n^{2}

Then, the mean of the new series is

\frac{1}{n} Σ y_{i} = \frac{1}{n} Σ (x_{i} + i^{2}) = \frac{1}{n} Σ x_{i} + \frac{1}{n} (1^{2} + 2^{2} + 3^{2} + . . . . + n^{2})

= \bar{X} + \frac{1}{n} \cdot \frac{n (n + 1) (2 n + 1)}{6} = \bar{X} + \frac{(n + 1) (2 n + 1)}{6}

Hence, the correct answer is option: 'B'.