Question

The sum of the first tern and the fifth term of an ascending A.P. is 26 and the product. of the second term by the fourth term is 160. And the sum of the first seven terms of this AP.

• A. 110
• B. 114
• C. 112
• D. 116

Answer 1

The correct option is C 112

Let the five terms of the A.P. be $a-2d,a-d,a,a+d,a+2d$.

Sum of the first and fifth terms is $(a-2d)+(a+2d)=2a$

$\Rightarrow$ $2a=26$

$\Rightarrow$ $a=13$

Product of second and fourth terms is $(a-d)(a+d)=a^2-d^2$

$\Rightarrow$ $a^2-d^2=160$

$\Rightarrow$ $169-d^2=160$

$\Rightarrow$ $d^2=9$

It is an ascending A.P. and hence $d$ takes positive values.$\Rightarrow$ $d=3$

Sum of $n$ terms in an A.P is $S_n=\frac{n}{2}(2a+(n-1\left)d\right)$

Here the first term is $a-2d$

Sum of seven terms=$S=\frac{7}{2}\left(2\right(13-2\left(3\right))+(7-1\left)\right(3\left)\right)$

$S=112$

$\therefore$ Option C is correct.