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LammettHash
I'm guessing S^20 is actually referring to the 20th partial sum [tex]S_{20}[/tex], with

[tex]S_{20}=1+(1+1\times0.1)+(1+2\times0.1)+\cdots+(1+18\times0.1)+(1+19\times0.1)[/tex]

in the arithmetic case, and

[tex]S_{20}=1+1.1+1.1^2+\cdots+1.1^{18}+1.1^{19}[/tex]

in the geometric case.

(a) Combining like terms, we have

[tex]S_{20}=20+0.1(1+2+\cdots+18+19)[/tex]

and invoking the formula

[tex]1+2+\cdots+(n-1)+n=\dfrac{n(n+1)}2[/tex]

we end up with

[tex]S_{20}=20+0.1\dfrac{19\times20}2=39[/tex]

(b) Multiply [tex]S_{20}[/tex] by 1.1, then subtract this from [tex]S_{20}[/tex]:

[tex]1.1S_{20}=1.1+1.1^2+1.1^3+\cdots+1.1^{19}+1.1^{20}[/tex]

[tex]S_{20}-1.1S_{20}=-0.1S_{20}=1-1.1^{20}[/tex]
[tex]\implies S_{20}=\dfrac{1.1^{20}-1}{0.1}\approx57.275[/tex]

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