1989 AHSME Problems/Problem 29

Problem

What is the value of the sum $S=\sum_{k=0}^{49}(-1)^k\binom{99}{2k}=\binom{99}{0}-\binom{99}{2}+\binom{99}{4}-\cdots -\binom{99}{98}?$


(A) $-2^{50}$ (B) $-2^{49}$ (C) 0 (D) $2^{49}$ (E) $2^{50}$

Solution

By the Binomial Theorem, $(1+i)^{99}=\sum_{n=0}^{99}\binom{99}{j}i^n =$ $\binom{99}{0}i^0+\binom{99}{1}i^1+\binom{99}{2}i^2+\binom{99}{3}i^3+\binom{99}{4}i^4+\cdots +\binom{99}{98}i^{98}$.

Using the fact that $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$, and $i^{n+4}=i^n$, the sum becomes:

$(1+i)^{99}=\binom{99}{0}+\binom{99}{1}i-\binom{99}{2}-\binom{99}{3}i+\binom{99}{4}+\cdots -\binom{99}{98}$.

So, $Re[(1+i)^{99}]=\binom{99}{0}-\binom{99}{2}+\binom{99}{4}-\cdots -\binom{99}{98} = S$.

Using De Moivre's Theorem, $(1+i)^{99}=[\sqrt{2}cis(45^\circ)]^{99}=\sqrt{2^{99}}\cdot cis(99\cdot45^\circ)=2^{49}\sqrt{2}\cdot cis(135^\circ) = -2^{49}+2^{49}i$.

And finally, $S=Re[-2^{49}+2^{49}i] = -2^{49}$.

$\fbox{B}$

See also

1989 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 28
Followed by
Problem 30
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