Question

If the sum of the first ten terms of an A.P. is 4 times the sum of the first five terms, then the ratio of the first term to the common difference is:

(a) 1: 2 

(b) 2: 1 

(c) 1: 4 

(d) 4: 1

Answer 1

Answer : (a) 1 : 2 

Let a and d be the first term and common difference respectively of the A.P.

S₁₀ = \(\frac{10}{2}\) [2a + 9d]

S₅ = \(\frac{5}{2}\)[2a + 4d]

Given, S₁₀ = 4S₅ 

⇒ 5(2a + 9d) = 4 × \(\frac{5}{2}\) [2a + 4d] 

⇒ 10a + 45d = 20a + 40d 

⇒ 5d = 10a 

⇒ \(\frac{a}{d}\)= \(\frac{5}{10}\) = \(\frac{1}{2}\)

⇒ a : d = 1 : 2