Question

The ratio of the 11^th term to the 18^th term of an AP is 2 : 3. Find the ratio of the 5^th term to the 21^st term, and also the ratio of the sum of the first five terms to the sum of the first 21 terms.

Answer 1

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.