# If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero.

Given: 10 × a10 = 15 × a15

To show : a25 = 0

Consider 10 × a10 = 15 × a15

10 [a + (10 - 1)d] = 15 [a + (15 - 1)d]

10a + 90d = 15a + 210d

- 5 a = 120 d

a = - 24d ………………(1)

Now, a25 = a + (25 - 1)d

a25 = a + 24d

a25 = - 24d + 24d (from equation 1)

a25 = 0

Hence, proved.

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