Question

Find S^20 if the series 1 + 1.1 ..... is a (a) arithmetic (b) geometric

Answer 1

I'm guessing S^20 is actually referring to the 20th partial sum
`S_(20)`, with


`S_(20)=1+(1+1*0.1)+(1+2*0.1)+\cdots+(1+18*0.1)+(1+19*0.1)`

in the arithmetic case, and


`S_(20)=1+1.1+1.1^2+\cdots+1.1^(18)+1.1^(19)`

in the geometric case.

(a) Combining like terms, we have


`S_(20)=20+0.1(1+2+\cdots+18+19)`

and invoking the formula


`1+2+\cdots+(n-1)+n=\frac{n(n+1)}2`

we end up with


`S_(20)=20+0.1\frac{19*20}2=39`

(b) Multiply
`S_(20)` by 1.1, then subtract this from
`S_(20)`:


`1.1S_(20)=1.1+1.1^2+1.1^3+\cdots+1.1^(19)+1.1^(20)`


`S_(20)-1.1S_(20)=-0.1S_(20)=1-1.1^(20)`

`\implies S_(20)=(1.1^(20)-1)/(0.1)\approx57.275`