Question

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

Answer 1

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.

Answer 2

Step-by-step explanation:

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.

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