Question

If $nAMs$ between $2$ and $38$, then sum of the resulting progression is $200$, the value of $n$ is

• A. $10$
• B. $8$
• C. $9$
• D. $Noneofthese$

Answer 1

Answer: (B)

$8$

Let $a_1$,$a_2$,$a_3$,------$a_n$ be $n$ arithmetic mean between $2$ and $38$

Then $2$,$a_1$,$a_2$,$a_3$,------$a_n$ is an A.P with $a=2$ and $a_{n+2}=38$

Now, $a_{n+2}=38$

$\Rightarrow a+\left(n+2-1\right)d=38$

$\Rightarrow 2+\left(n+1\right)d=38$

$\Rightarrow \left(n+1\right)d=36$ ---(i)

$S_{n+2}=200$

$\Rightarrow \frac{n+2}{2}\left[2a+\left(n+1\right)d\right]=200$

$\Rightarrow \frac{n+2}{2}\left[4+36\right]=200$

$\Rightarrow n+2=10$

$\Rightarrow n=8$