Question

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

Answer 1

Solution:-
(a + 10d)/(a + 17d) = 2/3
3a + 30d = 2a + 34d
3a - 2a = 34 - 30 
a = 4d
a₅ = a + 4d
= 4d + 4d 
a₅ = 8d
a₂₁ = a + 20d
= 4d + 20d
a₂₁ = 24d
So, a₅ : a₂₁ = 8 : 24 = 1 : 3
The ratio of the sum of the first 5 terms to the sum of the first 21 terms.
S₅ = 5/2(8d + 4d)
= 60d/2
S₅ = 30d
S₂₁ = 21/2(8d + 20d)
= 588d/2
S₂₁ = 294d
So, 
S₅ : S₂₁ = 30d : 294d
 = 5 : 49
Answer.

Answer 2

Answer:

(a + 10d)/(a + 17d) = 2/3

3a + 30d = 2a + 34d

3a - 2a = 34 - 30  

a = 4d

a₅ = a + 4d

= 4d + 4d  

a₅ = 8d

a₂₁ = a + 20d

= 4d + 20d

a₂₁ = 24d

So, a₅ : a₂₁ = 8 : 24 = 1 : 3

The ratio of the sum of the first 5 terms to the sum of the first 21 terms.

S₅ = 5/2(8d + 4d)

= 60d/2

S₅ = 30d

S₂₁ = 21/2(8d + 20d)

= 588d/2

S₂₁ = 294d

So,  

S₅ : S₂₁ = 30d : 294d

= 5 : 49

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