If the ratio of the sums of the first 9 terms to the first 15 terms of an arithmetic progression is 9 : 25, then the ratio of the 9th term to the 15th term is: 

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AAI Junior Assistant (Fire Service) Official Paper (Held On: 15 Nov 2022 Shift 1)
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  1. 3 : 5
  2. 19 : 31
  3. 17 : 29
  4. 14 : 19

Answer (Detailed Solution Below)

Option 3 : 17 : 29
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Detailed Solution

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Given:

If the ratio of the sums of the first 9 terms to the first 15 terms of an arithmetic progression is 9 : 25

Formula used:

Sum of first n terms of an AP: Sn = n/2 [2a + (n-1)d]

Calculation:

S9 = 9/2 [2a + 8d]

S15 = 15/2 [2a + 14d]

Given that, S9 / S15 = 9/25

⇒ (9/2 [2a + 8d]) / (15/2 [2a + 14d]) = 9/25

⇒ (9 [2a + 8d]) / (15 [2a + 14d]) = 9/25

⇒ (2a + 8d) / (2a + 14d) = 3/5

⇒ 10a + 40d = 6a + 42d

⇒ a = d/2

9th term (T9) = a + 8d = (d/2 + 8d) = 17d/2

15th term (T15) = a + 14d = (d/2 + 14d) = 29d/2

⇒ T9 / T15 = (17d/2) / (29d/2)

⇒ 17 / 29

∴ The correct answer is option (3).

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