If the sum of the first ten terms of an A.P. is 4 times the sum of the

1.	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» 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

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»

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