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10-97.

Antonio and Luann are trying to explain their strategy for computing the sum of a geometric series. They decide to generate a sequence that will be easy to sum so they can convince their classmates that their strategy really works. They decide to use $t(n)=3·10^{n-1}$.

1. The sum of the first six terms of the series is easy to calculate. You can do it in your head. Write out the first six terms and calculate the sum.

$3 + 30 + 300 + 3,000 + 30,000 + 300,000 = 333,333$

2. Now explain Luann and Antonio’s strategy for calculating the sum and use your answer to part (a) to check your result.

Write the series $3 + 30 + ... + 300,000 = S\left(6\right)$, twice. Multiply one of them by $10$.
Subtract: $10S\left(6\right) − S\left(6\right) = 9S\left(6\right) = 2,999,997$. Divide by $9$ to get $333,333$.

3. Represent the first $n$ terms of the series using summation notation and then write an expression for the total sum of $n$ terms.

$\displaystyle\sum_{k=1}^{n}{(3\cdot10^{k-1})}$