Question

There are $n$ sets of observations given as $\left(1\right), \, \left(2 ,3\right), \, \left(4 ,5 , 6\right), \, \left(7 ,8 , 9 ,10\right),.....$ The mean of the $13^{t h}$ set of observations is equal to

• A. $70$
• B. $80$
• C. $75$
• D. $85$

Answer 1

Answer: (D)

In the $n^{t h}$ set of observation, the total observations are $‘n’$
and the first observation of $n^{t h}$ set is
$1+2+3+....+\left(n - 1\right)+1=\frac{\left(n - 1\right) n}{2}+1=a$
Sum of all the $n$ observations in the $n^{t h}$ set
$=\frac{n}{2}\left[2 a + \left(n - 1\right) d\right]$
$=\frac{n}{2}\left[2 \left\{\frac{\left(n - 1\right) n}{2} + 1\right\} + \left(n - 1\right)\right]$
$=\frac{n}{2}\left[n^{2} - n + 2 + n - 1\right]=n\left(\frac{n^{2} + 1}{2}\right)$
Mean $=\frac{n \frac{\left(n^{2} + 1\right)}{2}}{n}=\frac{n^{2} + 1}{2}$
Hence, for $n=13$
Mean $=\frac{1 3^{2} + 1}{2}=\frac{170}{2}=85$