Question

The ratio of the sum use of n terms of two A.P.’s is (7n + 1) : (4n + 27). Find the ratio of their m^th terms

Answer 1

Let a₁ , a₂ be the first terms and d₁ , d₂ the common differences of the two given A.P.’s .Then the sums of their n terms are given by

`S_n = \frac { n }{ 2 } [2_1 + (n – 1) d_1 ], and S_n' = \frac{ n }{ 2 } [2a_2 + (n – 1) d_2 ]`

`:.S_n/(S'_n)=(n/2[2a_1+(n-1)d_1])/(n/2[2a_2+(n-1)d_2])=(2a_1+(n-1)d_1)/(2a_2+(n-1)d_2`

It is given that

`S_n/S_n^'=\frac{7n+1}{4n+27}`

`=>(2a_1+(n-1)d_1)/(2a_2+(n-1)d_2)=(7n+1)/(4n+27)`

To find the ratio of the mth terms of the two given A.P.’s, we replace n
by (2m – 1) in equation (i). Then we get

`:.(2a_1+(2m-2)d_1)/(2a_2+(2m-2)d_2)=(7(2m-1)+1)/(4(2m-1)+27)`

`=>(a_1+(m-1)d_1)/(a_2+(m-1)d_2)=\frac{14m-6}{8m+23}`

Hence the ratio of the m^th terms of the two A.P.’s is (14m – 6) : (8m+ 23)