Question

If the ratio of the sum of n terms of two APs is (7n + 1) : (4n + 27), then find the ratio of their mth terms.

Answer 1

Answer:

the ratio is  (14m-6) : (8m+23)

Step-by-step explanation:

Given:-

Sₙ / Sₙ' =  (7n + 1) / (4n + 27) ------------------------(1)

Let a₁, a₂ be the first terms of two APs respectively and d₁, d₂ be the common difference of 2 APs respectively.

Sₙ = n/2 (2a₁+(n-1)d₁)

Sₙ' = n/2 (2a₂+(n-1)d₂)    

Therefore,

Sₙ / Sₙ' = n/2 (2a₁+(n-1)d₁) / n/2 (2a₂+(n-1)d₂)    

= (2a₁+(n-1)d₁) / (2a₂+(n-1)d₂) ------------------------(2)

Equation (1) is equal to equation (2)

(2a₁+(n-1)d₁) / (2a₂+(n-1)d₂) =  (7n + 1) / (4n + 27) ---------------------(3)

To find the ratio of the mth terms of the two given AP's, we replace n by (2m-1) in equation (3).

Therefore,

(2a₁+(2m-2)d₁) / (2a₂+(2m-2)d₂) =  (7(2m-1) + 1) / (4(2m-1) + 27)

(a₁+(m-1)d₁) / (a₂+(m-1)d₂) =  (14m-6) / (8m+23)

Hence, the ratio is  (14m-6) : (8m+23)