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Question
Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.
Solution
Let the common difference of the AP be d.
First term, a = 5
Now,
`a_1 + a_2 +a_3 +a_4 = 1/2 (a_5 +a_6 +a_7 +a_8)` (Given)
`a+(a +d ) +( a+2d ) +(a+3d) = 1/2 [(a+4d)+(a+5d) +(a+6d) +(a+7d)]`
`[ a_n = a+ (n-1) d]`
⇒ 4a + 6d = `1/2` ( 4a + 22d)
⇒ 8a + 12d = 4a =22d
⇒ 22d - 12 d= 8a + 4a
⇒ 10d = 4a
⇒`d= 2/5 a`
⇒ `d= 2/5 xx 5 = 2 ` (a=5)
Hence, the common difference of the AP is 2.
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