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Math Help - Quadratics - sequences and squares

  1. #1
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    Quadratics - sequences and squares

    Okay, I've got this question and I've spent about an hour trying to figure it out and it's doing my head in:

    Write a formula for the nth term of this quadratic sequence:
    7, 15, 27, 43, 63
    nth term = ??

    Write this expression in completed square form:
    x - 10x +2

    Hence solve
    x - 10x +2 = -5

    Ok, any help would be appreciated as I've tried many ways of doing it....
    Thanks =D
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  2. #2
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    Hello, Soriku Strife!

    Write a formula for the n^{th} term of this quadratic sequence:
    . . 7, 15, 27, 43, 63, . . .
    Here's one approach to this problem (bear with me, it's quite long).


    We write a general quadratic function: . f(n) \;=\;an^2 + bn + c
    . . and we must determine a,\,b,\,c.

    We have three "unknowns", so we need three equations.
    We'll use the first three terms of the sequence.

    The first term is 7. .That is: . f(1) = 7
    . . a\cdot1^2 + b\cdot1 + c \:=\:7\quad\Rightarrow\quad a + b + c \:=\:7\;\;{\color{blue}[1]}

    The second term is 15. .That is: . f(2) = 15
    . . a\cdot2^2 + b\cdot2 + c\:=\:15\quad\Rightarrow\quad 4a + 2b + c \:=\:15\;\;{\color{blue}[2]}

    The third term is 27. .That is: . f(3) = 27
    . . a\cdot3^2 + b\cdot3 + c \:=\:27\quad\Rightarrow\quad 9a + 3b + c \:=\:27\;\;{\color{blue}[3]}

    \begin{array}{ccccc}\text{Subtract {\color{blue}[1]} from {\color{blue}[2]}:} & 3a + b & = & 8 & {\color{blue}[4]} \\<br />
\text{Subtract {\color{blue}[2]} from {\color{blue}[3]}:} & 5a + b & = & 12 & {\color{blue}[5]}\end{array}

    Subtract [4] from [5]: . 2a \:=\:4\quad\Rightarrow\quad\boxed{ a \:=\:2}

    Substitute into [4]: . 3(2) + b \:=\:8\quad\Rightarrow\quad\boxed{ b \:=\:2}

    Substitute into [1]: . 2 + 2 + c \:=\:7\quad\Rightarrow\quad\boxed{ c \:=\:3}


    Therefore, the function is: . f(n) \;=\;2n^2 + 2n + 3

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  3. #3
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    Thanks, made more sense than the way I did it
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  4. #4
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    Hello again, Soriku Strife!

    Write this expression in completed square form: . x^2 - 10x + 2

    We have: . x^2 - 10x + 2

    To complete the square: take one-half of the x-coefficient and square it.
    Add and subtract this quantity.

    . . x^2 - 10x \,{\color{blue}+\,25} + 2 \,{\color{blue}-\,25}\;=\;(x^2-10x + 25) - 23 \;=\;(x-5)^2-23



    Hence, solve: . x^2 - 10x +2 \:= \:-5

    We have the equation: . (x-5)^2 - 23 \:=\:-5

    Add 23 to both sides: . (x - 5)^2 \:=\:18

    Take square roots: . x - 5 \;=\;\pm\sqrt{18}\quad\Rightarrow\quad x - 5\;=\;\pm3\sqrt{2}

    Add 5 to both sides: . \boxed{x \;=\;5 \pm3\sqrt{2}}

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