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 $x , \frac{5 y }{4}, z$ 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 $x , xr , xr ^2$
$\therefore x + xr + xr ^2=42 $
$\text { and } 2 \cdot \frac{5}{4}( xr )= x + xr ^2 \Rightarrow 5 r =2+2 r ^2$
$\Rightarrow 2 r ^2-5 r +2=0 \Rightarrow ( r -2)(2 r -1)=0$
$\therefore r =2 \text { or } r =\frac{1}{2}$
$\therefore x =6 \text { or } x =24$
$\therefore x _1=24, x _2=6$
$\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} $