Question

Find the geometric sum: 40 + 20 + 10 + … + 0.078125

Answer 1

`S=40+20+10+\cdots+0.078125`

`S=40\left(1+\frac12+\frac14+\cdots+\frac1{512}\right)`

`S=40\left(1+2^(-1)+2^(-2)+\cdots+2^(-9)\right)`


`2^(-1)S=40\left(2^(-1)+2^(-2)+2^(-3)+\cdots+2^(-10)\right)`


`\implies S-2^(-1)S=40\left(1-2^(-10)\right)`

`\implies\frac12S=40\left(1-\frac1{1024}\right)`

`\implies S=80\left(1-\frac1{1024}\right)=(5115)/(64)=79.921875`