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 $13^{th}$ set of observations is equal to

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

Answer 1

Answer: (D)

In the $n^{th}$ set of observation, the total observations are $‘n’$
and the first observation of $n^{th}$ 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 $n^{th}$ set
$=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$
$=\frac{n}{2}\left[2\left\{\frac{\left(n-1\right)n}{2}+1\right\}+\left(n-1\right)\right]$
$=\frac{n}{2}\left[n^2-n+2+n-1\right]=n\left(\frac{n^2+1}{2}\right)$
Mean $=\frac{n\frac{\left(n^2+1\right)}{2}}{n}=\frac{n^2+1}{2}$
Hence, for $n=13$
Mean $=\frac{13^2+1}{2}=\frac{170}{2}=85$