a function f(x) satisfies the equation f(x)=f(x-1)+f(x+1) for all values of x. Define f(1)=1 and f(3)=3; then f(2)=1+3=4. find f(1867)?

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  • 4 weeks ago
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    It is given that: f(x) = f(x - 1) + f(x + 1)

    Add two more consecutive relations:

    f(x + 1) = f(x) + f(x + 2) …… [1]

    f(x + 2) = f(x + 1) + f(x + 3) …… [2]

    [1] + [2]:  f(x) + f(x + 3) = 0 …… [3]

    Also:  f(x + 3) + f(x + 6) = 0 …… [4]

    [3] - [4]:

    f(x) - f(x + 6) = 0

    f(x + 6) = f(x)

    Similarly, f(x + 12) = f(x + 6)

    f(x + 18) = f(x + 12)

    …… and so on

    It can be deduced that f(x + 6n) = f(x)

    where n is an integer.

    f(1867)

    = f(1 + 6×311)

    = f(1)

    = 1

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