Question

If 10 times the 10^th term of an A.P. is equal to 15 times the 15^th term, show that 25^th term of the A.P. is zero.

Answer 1

Here, let us take the first term of the A.P. as a and the common difference as d

We are given that 10 times the 10^th term is equal to 15 times the 15^th term.

We need to show that 25^th term is zero.

So, let us first find the two terms.

So, as we know,

`a_n = a + (n - 1)d`

For 10 th term (n = 10)

`a_10 = a + (10 - 1)d`

= a + 9d

For 15 th term (n = 15)

`a_15 = a + (15 - 1)d`

= a + 14d

Now, we are given,

10(a + 9d) = 15(a + 14d)

Solving this we get

10(a + 9d) = 15(a + 14d)

Solving this we get

10a + 90d = 15a + 210d

90d - 210d = 15a - 10a

-120d = 5a

-24d = a .......(1)

Next, we need to prove that the 25^th term of the A.P. is zero. For that, let us find the 25^th term using n = 25,

`a_25 = a + (25 - 1)d`

= -24d + 24d   (using 1)

= 0

Thus the 25^th term of the given A.P is zero.

Hence proved.