Question

After inserting $n$ A.M's between $2$and $38$, the sum of the resulting progression is $200$.

The value of $n$ is

• A. $10$
• B. $8$
• C. $9$
• D. None of these

Answer 1

The correct option is B

$8$

The explanation for the correct option:

Step1. Given data:

First-term, $a=2$

Last term, $l=38$

The number of A.M's inserted between $a$ and $l$ = $n$ terms.

Then the number of terms inserted between $a$ and $l$ = $n+2$ terms

Sum of the progression, $S_n=200$

The formula for the sum of $n$terms of A.P, is ${\mathit{S}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{n}}{\mathbf{2}}\mathbf{(}\mathit{a}\mathbf{+}\mathit{l}\mathbf{)}$

Step2, Find the value of $n$:

Now, putting the value of $a,l$and $n$

$\Rightarrow$$S_{n+2}=\frac{n+2}{2}(2+38)$

$\Rightarrow$ $200=\frac{n+2}{2}\left(40\right)$

$\Rightarrow$ $400=(n+2)\left(40\right)$

$\Rightarrow$ $10=n+2$

$\Rightarrow$ $n=8$

Hence, option (B) is the correct answer.