Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of

Question

Suppose that all the terms of an arithmetic progression (A.P.) are natural numbers. If the ratio of the sum of the first seven terms to the sum of the first eleven terms is 6:11 and the seventh term lies in between 130 and 140, then the common difference of this A.P. is

Professor PC Thomas Classes · Answer

(9) : $\frac{S_{7}}{S_{11}}=\frac{6}{11} \Rightarrow \frac{7(2 a+6 d)}{11(2 a+10 d)}=\frac{6}{11}$\ $\Rightarrow \quad 7 a+21 d=6 a+30 d \Rightarrow a=9 d$\ Now, $130<a+6 d<140$\ i.e., $130<15 d<140$ Hence, $d=9$\