Let $a = \frac{1}{3^{23}} + 1$. For every integer $n \ge 3$, define $$ f(n)=\binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+\binom{n}{2}a^{n-3}-\cdots+(-1)^{n-1}\binom{n}{n-1}a^0. $$ If $f(2007)+f(2008)=9k$, where $k\in\mathbb N$, determine $k$.

Question

Accepted Answer

1987. The function f(n) can be recognized as the expansion of (1 - a)^n using the binomial theorem, evaluated at a specific value of a. By calculating f(2007) and f(2008), we find that their sum simplifies to a multiple of 9, specifically 9 times 1987, confirming that k equals 1987.

Let \(a = \frac{1}{3^{23}} + 1\). For every integer \(n \ge 3\), define \[ f(n)=\binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+\binom{n}{2}a^{n-3}-\cdots+(-1)^{n-1}\binom{n}{n-1}a^0. \] If \(f(2007)+f(2008)=9k\), where \(k\in\mathbb N\), determine \(k\).

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