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Math Help - sequence formula

  1. #1
    Senior Member slevvio's Avatar
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    sequence formula

    The sequence {bn} is defined by b1 = 2, b2 = 8, b3 = 8bn-1 - 7bn-2 (n ≥ 3). Find b3 and b4 and guess a formula for bn. Use induction to prove your conjecture.

    The first part of this is easy, b3 and b4 being 50 and 344 respectively, but what I am having trouble with is guessing a general formula and I was wondering if anybody could help me. I know how to prove it but I don't have anything to prove yet! Thanks a lot
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  2. #2
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    Think about powers of 7.
    Hint: 50 = 1 + 7^2
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  3. #3
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    Hello, slevvio!

    The sequence \{b_n\} is defined by: . b_1 \:= \:2,\;b_2 \:= \:8,\;b_3 \:=\:8b_{n-1} - 7b_{n-2},\;\; (n \geq 3)

    Find b_3 and b_4 and guess a formula for b_n.
    Use induction to prove your conjecture.

    There is an intricate algebraic procedure for determining the general term.
    But as I started it, I eyeballed the first few terms ... and saw a pattern!

    The first few terms are:

    . . \begin{array}{ccc}b_1 & = & 2 \\ b_2 & = & 8 \\ b_3 & = & 50 \\ b_4 & = & 344 \\ b_5 & = & 2402 \\ \vdots & & \vdots\end{array}

    Those values looked "kind of familiar" . . .
    Then I recalled the powers-of-7 (which I had memorized years ago):
    . . \begin{array}{ccc}7^0 & = & 1 \\ 7^1 & = & 7 \\ 7^2 & = & 49 \\ 7^3 & = & 343 \\ 7^4 & = & 2401 \\ \vdots & & \vdots\end{array}

    Get it? . . . Those terms are one more than a power-of-7.

    I think we've got it! . . . \boxed{b_n \;=\;7^{n-1} + 1}


    I tested my conjecture with the next term: . b_5\:=\:8(2402) - 7(344) \:=\:16808

    . . which just happens to be: . 7^5 + 1 . . . Yay!

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