Question

If 5 times the 5th term of an ap is equal to 8 times the 8th term show that its 13th term is zero

Answer 1

Proved below

Explanation:

Given that

`5a_(5)=8a_(8)` .......(1

as we know

`a_(5)= a_(1) + 4d`

`a_(8)= a_(1) + 7d`

where d is the common difference. So, equation (1) would become

`5(a_(1) + 4d)=8(a_(1) + 7d)`

`5a_(1) + 20d=8a_(1) + 56d`

Rearranging

`5a_(1) - 8a_(1)=-20d + 56d`

`- 3a_(1)=36d`

So,

`a_(1)=-12d` .....(2)

As we know

`a_(13)= a_(1) + 12d` ......(3)

So, substituting the value of a1 into equation (3)

`a_(13)= -12d + 12d`

`a_(13)= 0`

Hence Proved