# Question 19

\begin{align} & \text{If the sum of the first term of G}\text{.P is 3, and the sum of the second and} \\ & \text{the third term is }-6,\text{ find the sum of the first term and the common ratio} \\ & \text{(A) 5 (B) }-2\text{ (C) }-3\text{ }(D)\text{ }-5 \\\end{align}
\begin{align} & \text{The }{{n}^{th}}-\text{term of a G}\text{.P is given as }a{{r}^{n-1}} \\ & {{T}_{n}}=a{{r}^{n-1}} \\ & {{T}_{1}}=a,{{T}_{2}}=ar,\text{ }{{T}_{3}}=a{{r}^{2}} \\ & {{T}_{1}}+{{T}_{2}}=a+ar=3 \\ & {{T}_{1}}+{{T}_{2}}=a(1+r)=3---(i) \\ & {{T}_{2}}+{{T}_{3}}=ar+a{{r}^{2}}=-6 \\ & {{T}_{2}}+{{T}_{3}}=ar(1+r)=-6---(ii) \\ & \text{Divide equation (}ii\text{) by }(i) \\ & \frac{ar(1+r)}{a(1+r)}=\frac{-6}{3} \\ & r=-2 \\ & \text{Substitute }r=-2\text{ into equation (i)} \\ & a(1-2)=3 \\ & -a=3 \\ & a=-3 \\ & \text{Sum of first term and common ratio}=(-2-3)=-5 \\ & a+r=-2-3=-5 \\\end{align}