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Math Help - induction

  1. #1
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    induction

    demostrate that the exponentiation of the number
    12
    890625 is another number which its lasts numbers are 12890625.

    exemple: 12890625^2 = 166168212890625

    Why can i demostrate it by induction? i don't know what i have to write on the right side of the equation (12890625)^n = ?
    to start the indutcion

    thanks
    PD: i'm sorry if my english is not too good, it isn't my usual lenguage.
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  2. #2
    o_O
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    n = 12890625

    From your clue, notice that:
    \begin{array}{rcl}n^2 & = & 166168212890625 \\ n^2 & = & 1661682 \cdot 10^8 + 12890625 \\ {\color{red}n^2} & {\color{red}=} & {\color{red}10^8 m + n}  \end{array}

    So assume the statement is true for P_{k}, that is n^k last 8 digits are n i.e. n^k = 10^8 s + n. So it remains to show that it is true for n^{k+1}
    Multiply both sides by n:
    \begin{array}{rcl} n^k & = & 10^8 s + n \\ n^{k+1} & = & 10^8 sn +  n^2 \\ n^{k+1} & = & 10^8 sn + {\color{red}(10^8m + n)} \\ & \vdots &  \end{array}

    Can you finish?
    Last edited by o_O; September 27th 2008 at 03:19 PM.
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  3. #3
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    yes, now i've could finish. thanks
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