Question

The arithmetic mean of $5$ consecutive integers starting with ${}^{\prime }{s}^{\prime }$ is ${}^{\prime }{a}^{\prime }.$ Then the arithmetic mean of $9$ consecutive integers that start with $s+2$ is

• A. $2+a+s$
• B. $22+a$
• C. $2s$
• D. $a+4$

Answer 1

The correct option is D $a+4$

Let $5$ consecutive integers be $s,s+1,s+2,s+3,s+4$
Arithmetic mean of these numbers is $a.$
$\Rightarrow \frac{5s+10}{5}=a$
$\Rightarrow s+2=a\dots \left(1\right)$

Mean of $9$ consecutive integers starting with $s+2$ is
$\frac{9s+(2+3+4+\cdots +10)}{9}$
$=\frac{9s+(1+2+3+4+\cdots +10-1)}{9}$
$=\frac{9s-1+\frac{10\times 11}{2}}{9}$
$=s+6$
$=a-2+6$
$=a+4$ [From equation $\left(1\right)$]