An algebra problem by Vermouth Xu

Algebra Level 3

If f(x) = 4 x + 1 4 x + 2 \frac { { 4 }^{ x+1 } }{ { 4 }^{ x }+2 } , find f(1/9)+f(2/9)+f(3/9)+f(4/9)+f(5/9)+f(6/9)+f(7/9)+f(8/9)


The answer is 16.

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Vermouth Xu by Vermouth Xu , 23, China

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2 solutions

Vermouth Xu
Apr 2, 2015

Observation is the key here. You can find f(1/9) and f(8/9) may pair up since the sum of the inputs is 1. The same applies to f(2/9) and f(7/9) etc.

For something more general, we can attempt f(x) + f(1-x), hoping to find something useful. And it turns out very promising, as the sum is a constant, 4. Therefore we know each pair goes up to 4, and because there are four pairs, we can say the answer is 16.

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Razing Thunder
Jul 3, 2020 
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a=[]
for x in range(1,9):
    x=x/9
    b=(4**(x+1))/(4**x + 2)
    a.append(b)
print(sum(a))

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