The ratio of the sum of n terms of two A.P's is (7n + 1) : (4n + 27). Find the ratio of m terms.

Solution:

A series of numbers that has a common difference between any pair of consecutive numbers is called an arithmetic progression.

Let's find the ratio of their m terms.

For first AP, let first term = a₁ and common difference = d₁

For second AP, let first term = a₂ and common difference = d₂

Let a_m1 be the m^th term of first AP and a_m2 be the m^th term of second AP.

We need to find trhe ratio of a_m1  =  a₁ + (m - 1) d₁  and  a_m2  =  a₂ + (m - 1) d₂     

According to the question, we get

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

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

By dividing both numerator and denominator with 2, we get

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

The equation is in the form of numbers of terms in A. P.

Therefore, by replacing n = 2m - 1, we get

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

⇒ [ a₁ + (m - 1) d₁ ] / [ a₂ + (m - 1) d₂ ] = ( 14m - 7 + 1 ) / ( 8m - 4 + 27 )

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

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

⇒ a_m1 / a_m2 = ( 14m - 6 ) / ( 8m + 23 )

Thus, If the ratio of the sum of n terms of two A.P's is (7n + 1) : (4n + 27), then the ratio of m terms is given by a_{m1 :} a_m2 = ( 14m - 6 ) : ( 8m + 23 )

---

The ratio of the sum of n terms of two A.P's is (7n + 1) : (4n + 27). Find the ratio of m terms.

Summary:

The ratio of their m^th terms is ( 14m - 6 ) : ( 8m + 23 ).