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jmc

algebra senior

Problem

What is the value of the sum

\frac{2}{3} + \frac{2 ^{2}}{3 ^{2}} + \frac{2 ^{3}}{3 ^{3}} + \dots + \frac{2 ^{10}}{3 ^{10}}

? Express your answer as a common fraction.

Solution

This is the sum of the series

a_{1} + a_{2} + \dots + a_{10}

with

a_{1} = \frac{2}{3}

and

r = \frac{2}{3}

.

Thus,

S = \frac{a ( 1 - r ^{n} )}{1 - r} = \frac{2}{3} \cdot \frac{1 - ( \frac{2}{3} ) ^{10}}{1 - \frac{2}{3}} = \frac{2}{3} \cdot \frac{1 - \frac{1024}{59049}}{\frac{1}{3}} = \frac{2}{3} \cdot \frac{3}{1} \cdot \frac{58025}{59049} = \frac{2 \cdot 58025}{59049} = \frac{116050}{59049} .

Final answer

\frac{116050}{59049}