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.

Answer 1

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 – 12d = 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.