If n arithmetic mans are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

$\begin{array}{l}\mathrm{Common} \mathrm{difference} =\mathrm{d}=\frac{100-\mathrm{a}}{\mathrm{n}+1}\\ {\mathrm{A}}_1=\mathrm{a}+\mathrm{d}\\ {\mathrm{A}}_{\mathrm{n}}=100-\mathrm{d}\\ \Rightarrow given that \frac{{\mathrm{A}}_1}{{\mathrm{A}}_{\mathrm{n}}}=\frac{1}{7}\Rightarrow \frac{\mathrm{a}+\mathrm{d}}{100-\mathrm{d}}=\frac{1}{7}\\ \Rightarrow 7\mathrm{a}+8\mathrm{d}=100\\ \Rightarrow 7\mathrm{a}+8\left(\frac{100-\mathrm{a}}{\mathrm{n}+1}\right)=100 ...\left(1\right)\\ but \mathrm{a}=33-n ...\left(2\right)\end{array}$

Now, by Eq. (1) and (2) 

$7{\mathrm{n}}^2-132\mathrm{n}-667=0$

$\mathrm{n}=23\text{ and }\mathrm{n}=\frac{-29}{7}\text{ reject. }$