Find a short expression for the long sum

Question

First of all, I'm quite new here, so sorry if this is not asked in the correct place.
The sum is $$X={{100}\choose{1}}+3\cdot{{100}\choose{3}}+5\cdot{{100}\choose{5}}+...+97\cdot{{100}\choose{97}}+99\cdot{{100}\choose{99}}$$
I have noticed that I can get a much simpler sum:
$$X={{100}\choose{1}}+99\cdot{{100}\choose{99}}+3\cdot{{100}\choose{3}}+97\cdot{{100}\choose{97}}+5\cdot{{100}\choose{5}}+95\cdot{{100}\choose{95}}+...+49\cdot{{100}\choose{49}}+51\cdot{{100}\choose{51}}$$.
Since $${{100}\choose{k}}={{100}\choose{100-k}}$$, it follows that the sum equals to:
$$X=100\cdot{{100}\choose{1}}+100\cdot{{100}\choose{3}}+...+100\cdot{{100}\choose{49}}$$
or:
$$X=100\cdot\sum_{n=0}^{24}{{100}\choose{2n+1}}$$
I'm not sure how to continue. I couldn't find a short term for the last sum.

Answer 1

You have reduced the problem to one that involves a summation over only the odd binomial coefficients. If we make the problem more general, then the solution becomes easier to see. Suppose we want to evaluate:

$$\sum_{k=0}^{N}f(k)\binom{N}{k}$$

where $f(k)$ is a periodic function. In your case $f(k)= 1/2\left[1-(-1)^k\right]$. The binomial theorem can thus be applied directly using the expression for $f(k)$. This is always possible because you can always expand $f(k)$ in Fourier series involving powers of complex exponentials.

The result is thus:

$$\frac{1}{2}\sum_{k=0}^{N}\left[1-(-1)^k\right]\binom{N}{k} = 2^{N-1}$$

We can also tackle the original summation directly as follows. We ahve:

$$(1+u)^N = \sum_{k=0}^N\binom{N}{k}u^k$$

differentiating both sides w.r.t. $u$ and then multiplying by $u$, gives:

$$N u (1+u)^{N-1} = \sum_{k=0}^N k \binom{N}{k}u^{k}$$

Define $g(u) =N u (1+u)^{N-1} $. We then have that $1/2\left[g(1) - g(-1)\right]$ is given by the desired summation, so this evaluates to $N 2^{N-2}$

Answer 2

We obtain \begin{align*} \color{blue}{\sum_{n=0}^{24}\binom{100}{2n+1}}&=\frac{1}{2}\sum_{n=0}^{49}\binom{100}{2n+1}\tag{1}\\ &=\frac{1}{2}\left(1\cdot\sum_{n=0}^{49}\binom{100}{2n+1}+0\cdot \sum_{n=1}^{50}\binom{100}{2n}\right)\tag{2}\\ &=\frac{1}{2}\sum_{n=0}^{100}\frac{1-(-1)^n}{2}\binom{100}{n}\tag{3}\\ &=\frac{1}{4}\sum_{n=0}^{100}\binom{100}{n}-\frac{1}{4}\sum_{n=0}^{100}(-1)^n\binom{100}{n}\tag{4}\\ &=\frac{1}{4}\cdot 2^{100}-\frac{1}{4}\left(1+(-1)\right)^{100}\tag{5}\\ &\,\,\color{blue}{=2^{98}}\tag{6} \end{align*}

Comment:

• In (1) we extend the region to all odd $n$ between $0$ and $100$ using the symmetry $\binom{n}{k}=\binom{n}{n-k}$.

• In (2) we add zero times all the even summands between $0$ and $100$.

• In (3) we collect the sums.

• In (4) we split the sum (somewhat differently to (2)).

• In (5) we apply the binomial theorem to both sums.

• In (6) we do a final simplification.