Soroban
Posts:
77
Registered:
12/6/04
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how do i find the nth term?
Posted:
Jan 20, 2004 12:31 PM
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Hello, Laura!
>I'm really stuck on this sequence of numbers; 5, 20, 50, 100, 175,
280,
>and how to find the formula for the nth term. Please can anybody
enlighten me?!
Taking consecutive differences, we get: 15, 30, 50, 75, 105, ...
Taking differences again: 15, 20, 25, 30, ...
Taking differences again: 5, 5, 5, ...
The third differences are constant.
Hence, the function is a CUBIC of the form: f(n) = an^3 + bn^2 + cn +
d
We know the first four values of this function: f(1) = 5, f(2) = 20,
f(3) = 50, f(4) = 100
Plugging those values into the general form, we get:
f(4) = 64a + 16b + 4c + d = 100
f(3) = 27a + 9b + 3c + d = 50
f(2) = 8a + 4b + 2c + d = 30
f(1) = a + b + c + d = 5
Solving this system, we get: a = 5/6, b = 5/2, c = 5/3, d = 0
The function is: f(n) = (5/6)n^3 + (5/2)n^2 + (5/3)n = 5n(n + 1)(n +
2)/6
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