Q. 125.0( 6 Votes )

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Answer :

Given: 24th term is twice the 10th term


a24 = 2a10


To prove: a72 = 2a34


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 = 24:


a24 = a + (24 – 1)d


a24 = a + 23d


When n = 34:


a34 = a + (34 – 1)d


a34 = a + 33d ………(i)


When n = 72:


a72 = a + (72 – 1)d


a72 = a + 71d


According to question:


a24 = 2a10


a + 23d = 2(a + 9d)


a + 23d = 2a + 18d


a – 2a + 23d – 18d = 0


-a + 5d = 0


a = 5d


Now, a72 = a + 71d


a72 = 5d + 71d


a72 = 76d


a72 = 10d + 66d


a72 = 2(5d + 33d)


{ a = 5d}


a72 = 2(a + 33d)


a72 = 2a34 (From (i))


Hence Proved


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