#### Question

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

#### Solution

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 the term of the given A.P is zero

Hence proved