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Math (sequence)

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Find the first four terms of each sequence.

A1= 2
A2=5
An=An-1 + An-2, n≥3

  • Math (sequence) - ,

    This is a recursive relationship very similar to the Fibonacci numbers, except that A1=2 and A2=5, whereas F1=1 and F2=1 for Fibonacci numbers.

    We start with
    An=An-1+An-2
    Substitute An-1=An-2+An-3, we get
    An=2An-2+An-3
    Similarly, and continuing,
    An=3An-3+2An-4
    An=5An-4+3An-5
    An=8An-5+5An-6
    An=11An-6+8An-5
    ....
    Replacing the coefficients by Fibonacci numbers, where
    F0=0, F1=1, F2=1, F3=2, F4=3...
    and continuing the substitution until A2 and A1, we would get:
    An=Fn-2*A1+Fn-1*A2

    Thus
    A4=F2*A1+F3*A2
    =1*2+2*5
    =12

    A8
    =F6*2+F7*5
    =8*2+13*5
    =16+65
    =81

    Work out A8 manually as a check.

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