Sum of first n terms of two different a.p.s are in ratio(3n+13)(7n-13)

1.	Sum of first n terms of two different a.p.s are in ratio(3n+13)(7n-13) find the ratio 5th terms.
Answer» GIVEN :- Sum of n terms of two different APs are in the ratio 3n + 13 : 7n - 13 To Find :- Ratio of 5th terms. Solution :- Given us the sum of n terms of two different APs as, ⇒ S₁ / S₂ = (5n + 13) / (7N - 13) Let the FIRST term and the COMMON difference of the two APs be a, a₂ and d, d₂ respectively. Also, We know ⇒ Sum of n terms of an AP = n / 2 { 2a + (n - 1)d } So, ⇒ [ n/2 { 2a + (n - 1)d } ] / [ n/2 { 2a₂ + (n - 1)d₂ ] = (5n + 13) / (7n - 13) ⇒ { 2a + (n - 1)d } / { 2a₂ + (n - 1)d₂ } = (5n + 13) / (7n - 13) So, Here we have to substitute a value of n for which the EQUATION becomes the ratio of the 5th elements of the APs. n = 8 ⇒ { 2a + (9 - 1)d } / { 2a₂ + (9 - 1)d₂ } = (5×9 + 13) / (7×9 - 13) ⇒ a + 4d / a₂ + 4d₂ = 58 / 50 ⇒ a₄ / a’₄ = 29 / 25 Hence, The ratio of their fifth terms will be 29 : 25.

Sum of first n terms of two different a.p.s are in ratio(3n+13)(7n-13) find the ratio 5th terms.

Answer»

Given us the sum of n terms of two different APs as,

⇒ S₁ / S₂ = (5n + 13) / (7N - 13)

Let the FIRST term and the COMMON difference of the two APs be a, a₂ and d, d₂ respectively.

Also, We know

⇒ Sum of n terms of an AP = n / 2 { 2a + (n - 1)d }

So,

⇒ [ n/2 { 2a + (n - 1)d } ] / [ n/2 { 2a₂ + (n - 1)d₂ ] = (5n + 13) / (7n - 13)

⇒ { 2a + (n - 1)d } / { 2a₂ + (n - 1)d₂ } = (5n + 13) / (7n - 13)

So, Here we have to substitute a value of n for which the EQUATION becomes the ratio of the 5th elements of the APs.

n = 8

⇒ { 2a + (9 - 1)d } / { 2a₂ + (9 - 1)d₂ } = (5×9 + 13) / (7×9 - 13)

⇒ a + 4d / a₂ + 4d₂ = 58 / 50

⇒ a₄ / a’₄ = 29 / 25

Hence,

The ratio of their fifth terms will be 29 : 25.