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Math Help - Sequence, sum of elements

  1. #1
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    Sep 2011
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    Sequence, sum of elements

    Hi!

    Could you help me with this assignment?

    What is the sum of n first elements of the sequence (a_n)where:

    a_1=2, a_2=22, a_3=222, ... ?

    Thanks.
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  2. #2
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    Mar 2011
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    Awetuouncsygg
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    Re: Sequence, sum of elements

    First you should have tried to find a formula for a_{n}. It's easy to observe that:

    a_{n}=2(10^{n-1}+10^{n-2}+...+10^{1}+10^{0})=2\cdot \frac{10^{n}-1}{10-1}=\frac{2}{9}\cdot (10^{n}-1)

    Now that you "know" a_{n} it's simpler to calculate the sum:

    a_{1}+a_{2}+...+a_{n-1}+a_{n}=\frac{2}{9}\left ( 10^{1}+10^{2}+...+10^{n-1}+10^{n} -n\right )=\frac{2}{9}\left ( 10\cdot \frac{10^{n}-1}{10-1} -n \right )

    S_{n}=\frac{2}{9}\cdot \frac{10^{n+1}-10-9n}{9}=\frac{2}{81}\cdot \left (10^{n+1}-9n-10  \right )

    Pretty clear I hope...
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