Nested Function Inversion
If
\[f(n + 1) = (-1)^{n + 1} n - 2f(n)\]for $ n \ge 1, $ and $ f(1) = f(1986), $ compute
\[f(1) + f(2) + f(3) + \dots + f(1985).\]
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
We can list the equations
\begin{align*}
f(2) &= 1 - 2f(1), \\
f(3) &= -2 - 2f(2), \\
f(4) &= 3 - 2f(3), \\
f(5) &= -4 - 2f(4), \\
&\dots, \\
f(1985) &= -1984 - 2f(1984), \\
f(1986) &= 1985 - 2f(1985).\end{align*}Adding these equations, we get
\[f(2) + f(3) + \dots + f(1986) = (1 - 2 + 3 - 4 + \dots + 1983 - 1984 + 1985) - 2f(1) - 2f(2) - \dots - 2f(1985).\]To find $ 1 - 2 + 3 - 4 + \dots + 1983 - 1984 + 1985, $ we can pair the terms
\begin{align*}
1 - 2 + 3 - 4 + \dots + 1983 - 1984 + 1985 &= (1 - 2) + (3 - 4) + \dots + (1983 - 1984) + 1985 \\
&= (-1) + (-1) + \dots + (-1) + 1985 \\
&= -\frac{1984}{2} + 1985 \\
&= 993.\end{align*}Hence,
\[f(2) + f(3) + \dots + f(1986) = 993 - 2f(1) - 2f(2) - \dots - 2f(1985).\]Then
\[2f(1) + 3f(2) + 3f(3) + \dots + 3f(1985) + f(1986) = 993.\]Since $ f(1986) = f(1), $
\[3f(1) + 3f(2) + 3f(3) + \dots + 3f(1985) = 993.\]Therefore, $ f(1) + f(2) + f(3) + \dots + f(1985) = \boxed{331} $.
Freeflows