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

Given,

10 times the 10th term of an A.P. is equal to 15 times the 15th term

10a10 = 15a15

To prove: a25 = 0

We know, an = a + (n – 1)d where a is first term or a1 and d is common difference and n is any natural number

When n = 10:

a10 = a + (10 – 1)d

a10 = a + 9d

When n = 15:

a15 = a + (15 – 1)d

a15 = a + 14d

When n = 25:

a25 = a + (25 – 1)d

a25 = a + 24d ………(i)

According to question:

10a10 = 15a15

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

10a + 90d = 15a + 210d

10a – 15a + 90d – 210d = 0

-5a – 120d = 0

-5(a + 24d) = 0

a + 24d = 0

a25 = 0 (From (i))

Hence Proved

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