Question

Sum of all terms of an infinite G.P. is 1/5 times the sum of odd terms. The common ratio is ...​

Answer 1

Answer:

Step-by-step explanation:

Sun of all terms = a/1-r

Sun of odd terms = (a, ar², ar⁴,...)= a/1-r²

Now equate as

a/1-r =(1/5) *a/1-r²

R= -4/5 also r ≠1

Answer 2

The common ratio is$\frac{-4}{5}$

Step-by-step explanation:

The infinite G.P. series of all terms is $a, ar , ar^{2}, ar^{3}....$

The infinite G.P. series of odd terms is $a, ar^{2}, ar^{4}, ar^{6}....$

The sum of all terms of infinite G.P. series is $\frac{a}{1-r}$

The sum of odd terms of infinite G.P. series is $\frac{a}{1-r^{2} }$

Given that,

Sum of all terms of an infinite G.P. =  $\frac{1}{5} \times$ the sum of odd terms

    $\frac{a}{1-r} = \frac{1}{5} \times \frac{a}{1-r^{2} }$

    $5 \times \left ( 1-r^{2} \right ) = 1-r$

    $5r^{2}-r-4$

    $\left ( r-1 \right )\left ( 5r+4 \right )$

    $r=1  , r= \frac{-4}{5}$}

    $r\neq 1$

The common ratio is$\frac{-4}{5}$