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vovangra [49]
3 years ago
13

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

Mathematics
1 answer:
quester [9]3 years ago
7 0
I'm guessing S^20 is actually referring to the 20th partial sum S_{20}, with

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

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=\dfrac{n(n+1)}2

we end up with

S_{20}=20+0.1\dfrac{19\times20}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}=\dfrac{1.1^{20}-1}{0.1}\approx57.275
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