Q. 104.5( 4 Votes )

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.

Answer :

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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