Question
Download Solution PDFIf 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:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
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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