The sum of the first term and the sixth term of an A.P. is 28 and the product of the second term and the fourth term is 160. What is the sum of the first six terms of the A.P.?

189

Let the five terms of the A.P. :- 

 a−2d, a−d, a, a+d, a+2d
Sum of the first and fifth terms is (a−2d)+(a+2d)=2a

2a= 28

a=14

Product of second and fourth terms is (a−d)(a+d)

= a² - d²

ATQ,

a² - d² = 160
196 - d² = 160

 d²  = 9

Is is an ascending AP . So d must be positive .

So, d = 3

Sum of n terms in an A.P is =n/2(2a+(n−1)d)

Here the first term is a−2d

Sum of 6 terms=

S=6/2 ( 2( 14 - 6 ) + (6 - 1 ) )3

= 3 ( 16 + (6 - 1 ) )3

=189

The correct answer is Option (2) → 189