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Math Help - Conjuncture a Formula and Probe it Using Induction

  1. #1
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    Conjuncture a Formula and Probe it Using Induction

    Hey all. I have a problem in my stage 2 pure maths paper that I'm not sure how to solve. Here it is:

    Let x(1), x(2), x(3),... be a sequence of integers defined recursively by :
    x(1) = 3, x(2) = 18, and x(n) = 6x(n-1) - 9x(n-2)
    for integers n >= 3. Conjecture a formula for x(n) and prove it using mathematical induction.


    I'm really not sure how to start it... I think I will be fine with the induction but am not sure how to get the formula...
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  2. #2
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    Hello, mrtwigx!

    Finding the formula can be quite a task . . . or quite simple.


    Let x(1),\:x(2),\:x(3)\:\hdots be a sequence of integers defined recursively by:

    . . x(1) = 3,\;\;x(2) = 18,\;\;x(n) \:=\: 6\!\cdot\!x(n-1) - 9\!\cdot\!x(n-2)\;\text{ for }n \geq 3

    Conjecture a formula for x(n) and prove it using mathematical induction.
    Crank out the first few terms of the sequence and hope for a pattern:

    . . \begin{Bmatrix}x(1) &=& 3 &=& 1\cdot 3\\ x(2) &=& 18 &=& 2\cdot3^2\\ x(3) &=& 81 &=& 3\cdot3^2\\ x(4) &=& 324 &=& 4\cdot 3^4 \\x(5) &=& 1215 &=& 5\cdot3^5 \end{Bmatrix}\qquad \Leftarrow\:\text{ There!}


    The n^{th} term seems to be: . x(n) \;=\;n\!\cdot\!3^n


    Go for it!

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