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in Arithmetic Progression by (22.8k points)
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n arithmetic means are inserted between 3 and 17. If the ratio of the last and the first arithmetic mean is 3: 1, then n is equal to

(a) 5 

(b) 6 

(c) 7 

(d) 9

1 Answer

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Best answer

Answer : (b) 6

Let a1, a2, a3, ..... , an be the n arithmetic means between 3 and 17. 

Then, 3, a1, a2, a3, ..... , an, 17 form an A.P.

Let d be the common difference of this A.P. 

Then, a1 = 3 + d and an = 17 – d

Given, 

\(\frac{a_n}{a_1} \) = \(\frac{3}{1}\) ⇒ \(\frac{17-d}{3+d}\) = \(\frac{3}{1}\) 

⇒ 17 – d = 9 + 3d 

⇒ 4d = 8 ⇒ d = 2. 

Also 17 is the (n + 2)th term of the given A.P. 

∴ 17 = 3 + (n + 2 – 1)2 

⇒ 17 = 3 + (n + 1)2 

⇒ 14 = (n + 1)2 

⇒ n + 1 = 7 

n = 6.

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