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Question 6
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.
Given that, a11:a18=2:3
a+10da+17d=23
3a+30d=2a+34d
a=4d ...(i)
Now, a5 = a + 4d = 4d + 4d = 8d [from Eq.(i)]
And a21 = a + 20d = 4d + 20d = 24d [from Eq. (i)]
a5:a21 = 8d : 24d = 1 : 3
Now, sum of the first five terms, S5=52[2a+(51)d]
=52[2(4d)+4d] [from Eq.(i)]
=52(8d+4d)=52×12d=30d
And, sum of the first 21 terms, S21=212[2a+(211)d]
=212[2(4d)+20d] from Eq..(i)]
So, ratio of the sum of the first five terms to the sum of the first 21 terms is,
S5:S21=30d:294d=5:49

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