Question

The ratio of the 11^th term to 17^th term of an A.P. is 3 : 4. Find the ratio of 5^th term to 21^st term of the same A.P. Also, find the ratio of the sum of first 5 terms to that of first 21 terms.

Answer 1

Given: The 11^th term : 17^th term = 3 : 4 for an A.P. Let first term = a and common difference = d.

Step-wise calculation:

1. Express the terms:

T₁₁ = a + 10d

T₁₇ = a + 16d

2. Use the given ratio:

(a + 10d) : (a + 16d) = 3 : 4

⇒ 4(a + 10d) = 3(a + 16d)

⇒ 4a + 40d = 3a + 48d

⇒ a = 8d

3. Ratio of 5^th to 21^st term:

T₅ = a + 4d

T₂₁ = a + 20d

Substitute a = 8d : T₅ : T₂₁ 

= (8d + 4d) : (8d + 20d)

= 12d : 28d

= 3 : 7

4. Ratio of sum of first 5 terms to first 21 terms:

`S_5 = 5/2 [2a + 4d]` 

`S_21 = 21/2 [2a + 20d]` 

So S₅ : S₂₁ = 5(2a + 4d) : 21(2a + 20d).

Substitute a = 8d (so 2a = 16d):

S₅ : S₂₁ = 5(16d + 4d) : 21(16d + 20d)

= 5 × 20d : 21 × 36d 

= 100d : 756d

= 25 : 189