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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