Paragraph: Let P be the sum of first n positive terms of an increasing arithmetic progression A . Let Q be the sum of first n positive terms of another increasing arithmetic progression B . Let P: Q=(5 n+4):(9 n+6) Question: If d is the common difference of A , and D is the common difference of B , then which one of the following is always correct?

Step-by-step solution

There are two AP with the different first terms and common difference. For the first AP A: Let the first term be a and common difference = d Sum of n terms P = n 2 [2 a+(n-1) d] and n ^ th term a_n=a+(n-1) d For the second AP B: Let the first term be M and common difference = D Sum of n terms Q = n 2 [2 M+(n-1) D] and n^ th term M_n=M+(n-1) D It is given that aligned & Sum of n terms of first A.P Sum of n terms of second A.P = 5 n+4 9 n+6 & n 2 [2 a+(n-1) d] n 2 [2 M+(n-1) D] = 5 n+4 9 n+6 & [2 a+(n-1) d] [2 M+(n-1