Question

The ratio of the 11^th term to the 18th 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, `a_5 = a + 4d = 4d + 4d = 84d`  ......[From equation (i)]

And `a_21 = a + 20d = 4d + 20d = 24d`  .....[From equation (i)]

∴ `a_5 : a_21 = 8d : 24d = 1 : 3`

Now, sum of the first five terms,

`S_5 = 5/2[2a + (5 - 1)d]`  .....`[because 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,

`S_21 = 21/2[2a + (21 - 1)d]`

= `21/2[2(4d) + 20d]`  ....[From equation (i)]

= `21/2(28d)`

= 294d

So, ratio of the sum of the first five terms to the sum of the first 21 terms is

S₅ : S₂₁ = 30d : 29d = 5:49