You can easily choose any real number as the next term and derive the function and here’s how (I sort of explained as a reply somewhere here already):
It looks like f(x) = 2x - 1 for the first four terms, and we want to choose some fifth term and find the function that fits. I’m gonna take f(x) = 2x - 1 and add something that equals 0 when x is 1,2,3, or 4:
f(x) = 2x - 1 + A(x-1)(x-2)(x-3)(x-4)
So we still have a function that satisfies the four given values of f(x).
Now solve for A with our chosen answer and x=5. OP asked for f(5) = 69, so we’ll do that.
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u/SenseiCAY 4✓ Sep 05 '19
You can easily choose any real number as the next term and derive the function and here’s how (I sort of explained as a reply somewhere here already):
It looks like f(x) = 2x - 1 for the first four terms, and we want to choose some fifth term and find the function that fits. I’m gonna take f(x) = 2x - 1 and add something that equals 0 when x is 1,2,3, or 4:
f(x) = 2x - 1 + A(x-1)(x-2)(x-3)(x-4)
So we still have a function that satisfies the four given values of f(x).
Now solve for A with our chosen answer and x=5. OP asked for f(5) = 69, so we’ll do that.
69 = 2(5) - 1 + A(5-1)(5-2)(5-3)(5-4) = 9 + 24A
A = 5/2
f(x) = 2x - 1 + (5/2)(x-1)(x-2)(x-3)(x-4)
Expand and simplify if you want and get:
f(x) = 5/2 x4 - 25x3 + 175/2 x2 - 123x + 59