The ratio of the sums of the first n terms of two arithmetic progressions is given by 3n+2 2n+7 for all n 1 . If the second term of the first arithmetic progression is 33 , then the fifth term of the second arithmetic progression is equal to

Step-by-step solution

Let the sum of the first n terms of the two A.P.s be S_n and T_n . Given S_n T_n = 3n+2 2n+7 . Since the sum of n terms of an A.P. is of the form An^2 + Bn (with no constant term), we can multiply the numerator and denominator by n to represent the sums properly: S_n T_n = n(3n+2) n(2n+7) Thus, we can write S_n = c(3n^2 + 2n) and T_n = c(2n^2 + 7n) for some non-zero constant c . The n^ th term of the first A.P. is a_n = S_n - S_ n-1 . a_n = c[3n^2 + 2n - (3(n-1)^2 + 2(n-1))] a_n = c[3n^2 + 2n - (3n^2 - 6n + 3 + 2n