Question

The first three terms of a geometric sequence is $x,y,z$ and these have the sum equal to 42 . If the middle term y is multiplied by $\frac{5}{4}$, then numbers $x,\frac{5y}{4},z$ now form an A.P. If the largest possible value and smallest possible value of $x$ are $x_1,x_2$ then

• A. sum of infinite terms of G.P with $x_1,x_2$ as first two terms is 32 .
• B. sum of infinite terms of GP with $x_1,x_2$ as first two terms is 18 .
• C. $x_1+x_2=30$
• D. ratio of A.M. and GM. of $x_1,x_2$ is $\frac{5}{4}$

Answer 1

Answer: (A), (C), (D)

Let terms $x,y,z$ be $x,xr,x{r}^2$
$\therefore x+xr+x{r}^2=42$
$\text{ and }2\cdot \frac{5}{4}(xr)=x+x{r}^2\Rightarrow 5r=2+2r^2$
$\Rightarrow 2r^2-5r+2=0\Rightarrow (r-2)(2r-1)=0$
$\therefore r=2\text{ or }r=\frac{1}{2}$
$\therefore x=6\text{ or }x=24$
$\therefore x_1=24,x_2=6$
$\text{ sum of infinite G.P. }=\frac{24}{1-(1/4)}=32$
$\text{ and }{x}_1+x_2=30$
$\text{ and }\frac{\text{ A.M. }}{\text{ G.M. }}=\frac{30/2}{\sqrt{24\cdot 6}}=\frac{15}{12}=\frac{5}{4}$