Question

sum of first two terms of GP s 4/3 and next two terms is 1/2 find the GP

Answer 1

$$\begin{gathered} LetheGPbea,ar,a{r}^2,a{r}^3........a{r}^n.wherer(\ne 0)isthecommonratio \\ Sumoffirsttwoterms=\frac{4}{3} \\ \therefore a+ar=\frac{4}{3}..........\left(1\right) \\ Sumofnexttwoterms=\frac{1}{2} \\ \therefore a{r}^2+a{r}^3=\frac{1}{2}.........\left(2\right) \\ Dividing\left(2\right)by\left(1\right),weget \\ \frac{a{r}^2+a{r}^3}{a+ar}=\frac{{\displaystyle \frac{1}{2}}}{{\displaystyle \frac{4}{3}}} \\ \Rightarrow \frac{a{r}^2(1+r)}{a(1+r)}=\frac{3}{8} \\ \Rightarrow r^2=\frac{3}{8} \\ \Rightarrow r=\sqrt{\frac{3}{8}}=0.612 \\ Hencea+a\times 0.612=\frac{4}{3} \\ \Rightarrow 1.612a=\frac{4}{3} \\ \Rightarrow a=0.827 \\ \therefore ar=0.506,a{r}^2=0.309,a{r}^3=0.189 \\ SotheGPis0.827,0.506,0.309,0.189 \end{gathered}$$