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Question
The ratio of the 11th term to the 18th term of an AP is 2 : 3. Find the ratio of the 5th term to the 21st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.
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Solution
Let a and d be the first term and common difference of an AP respectively.
Given that, a11 : a18 = 2 : 3
⇒ `(a + 10d)/(a + 17d) = 2/3`
⇒ 3a + 30d = 2a + 34d
⇒ a = 4d ...(i)
Now, a5 = a + 4d
= 4d + 4d
= 84d ...[From equation (i)]
And a21 = a + 20d
= 4d + 20d
= 24d ...[From equation (i)]
∴ a5 : a21 = 8d : 24d = 1 : 3
Now, sum of the first five terms,
S5 = `5/2[2a + (5 - 1)d]` ...`[∵ S_n = n/2[2a + (n - 1)d]]`
= `5/2[2(4d) + 4d]` ...[From equation (i)]
= `5/4(8d + 4d)`
= `5/2 xx 12d`
= 30d
And sum of the first 21 terms,
S21 = `21/2[2a + (21 - 1)d]`
= `21/2[2(4d) + 20d]` ...[From equation (i)]
= `21/2(28d)`
= 294d
S5 : S21 = 30d : 294d = 5 : 49
So, ratio of the sum of the first five terms to the sum of the first 21 terms is 5 : 49.
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