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

Given: 10 × a10 = 15 × a15

To Prove: a25 = 0

Now,

10 × (a + 9d) = 15 × (a + 14d)

10a + 90d = 15a + 210d

10a – 15a = 210d – 90d

–5a = 120d

a = –24d …(i)

Now,

an = a + (n – 1)d

a25 = –24d + (25 – 1)d [from (i)]

a25 = –24d + 24d

a25 = 0

Hence Proved

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