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Question
Find the common difference of an A.P. whose first term is 5 and the sum of first four terms is half the sum of next four terms.
Solution
Let a and d be the first term and common difference of A.P. respectively.
Given, a = 5
a1 + a2 + a3 + a4 = `(1)/(2)` (a5 + a6 + a7 + a8)
∴ a + (a + d) + (a + 2d) + (a + 3d)
= `(1)/(2)[(a + 4d) + (a + 5d) + (a + 6d) + (a + 7d)]`
⇒ 2(4a + 6d) = (4a + 22d)
⇒ 2(20 + 6d) = (20 + 22d) ...(∵ a = 5)
⇒ 40 + 12d = 20 + 22d
⇒ 10d = 20
⇒ d = 2
Thus, the common difference of A.P. is 2.
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