Question

let sum of infinite geometric progression with non zero common ratio is 4 then the sum of all possible values of its first term is

Answer 1

Answer with explanation:

→Sum of Infinite Geometric Progression with non zero common ratio=4

→It is not given either,  Common ratio > 1 or Common ratio <1, or Equal to 1.

→So, the Geometric Sequence having first term equal to ,a, and common ratio=r,will be as Follows:→a, a r,a r², a r³,......

→Sum to Infinity is given by

$\rightarrow S_{\infty}=\frac{a}{1-r}\\\\4=\frac{a}{1-r}\\\\a=4-4 r\\\\a +4 r=4\text{as}S_{\infty}=\frac{a}{1-r}\text{for},r<1\\\\\rightarrow S_{\infty}=\frac{a}{r-1}\\\\4=\frac{a}{r-1}\\\\a=-4+4 r\\\\a -4 r+4=0\\\\\text{as}S_{\infty}=\frac{a}{r-1}\text{for},r>1$

→The two equation in terms of , a and r, is

a + 4 r=4, for, r<1

a -4 r = -4, for , r>1.

→Also,for, r≠ 0, and r≠1,that is, for, r=0, a=0, and for, r=1,a=0, then the Sequence will not exist.

→So,when you plot these two lines in coordinate plane, for, r<1, there will be infinite values of a, as it is not given that which kind of values it can take , that is either only integral or rational, values it can take.

→So, there are infinite real values of a for, r<1 and , r>1, and to find it's sum is not possible.