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A function $f(x)$ is defined so that$$ f(x) = f(x - 2) - x(x + 2) $$for all the integer values of $x$ and $f(1) + f(4) = 0$. What is the value of

$$f(1) + f(2) + f(3) + f(4) + f(5) + f(6)?$$

• 0

• 89

• -89

• None of these

This is a practice question from here, not any test/HW question.

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2 Answers

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Just define $f(1)=k$. You are then given $f(4)=-k$. Use the recursion to find the other four values in terms of $k$. When you add them up, the $k$s disappear.

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Let $S = f(1) + f(2) + f(3) + f(4) + f(5) + f(6)$

As $f(1) + f(4) = 0$, therefore $S = f(2) + f(3) + f(5) + f(6)$ : (1)

$f(2) = f(0) - 8$

$f(3) = f(1) - 15$

$f(4) = f(2) - 24 = f(0) - 32$

$f(5) = f(3) - 35 = f(1) - 50$

$f(6) = f(4) - 48 = f(0) - 80$

Put the above values in equation (1), we get

$S = f(0) - 8 + f(1) - 15 + f(1) - 50 + f(0) - 80$

$S = 2(f(0) + f(1)) - 153$ : (2)

As we already know $f(1) + f(4) = 0 \Rightarrow f(1) + f(0) - 32 = 0 \Rightarrow f(1) + f(0) = 32$

Putting this value in equation (2), we get $S = 2(32) - 153 = -89$

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