<s>[INST] Evaluate the following $(3-2)(5-3)(7-5)........(1995-1993)$ 
Evaluate  the following 
  $(3-2)(5-3)(7-5)....(1995-1993)$

I have problem in counting the terms since the answer will be $2^n$ , $n$ is the number of terms. 
So I count them as follows:
$1995=2n-1$
$1996=2n \to\ n= 998$ ; but since the first term equals $1$
The the answer will be  $2^{997}$
Is my answer right? 
 [/INST] Since the first factor is indeed $3-2=1$, we will simply ignore it and start by counting $5-3$ as factor #1.
Factor $\#\color{red}{1}$ is $5-3$, which is $(2 \cdot \color{red}{2} + 1) - (2 \cdot \color{red}{2} - 1)$.
Factor $\#\color{red}{2}$ is $7-5$, which is $(2 \cdot \color{red}{3} + 1) - (2 \cdot \color{red}{3} - 1)$.
Factor $\#\color{red}{3}$ is $9-7$, which is $(2 \cdot \color{red}{4} + 1) - (2 \cdot \color{red}{4} - 1)$.
See the pattern?
Factor $\#\color{red}{n}$ is $(2 \cdot \color{red}{[n+1]} + 1) - (2 \cdot \color{red}{[n+1]} - 1)$.
Which $n$ gives us $2[n+1] + 1 = 1995$ and $2[n+1] - 1 = 1993$?  It's the same $n$ for both so we only need to solve one of them.  Either way we get $n = 996$.
So yes, there are $997$ total factors.  One of them is $1$ and the other $996$ are $2$, so the answer is $1 \cdot 2^{996} = 2^{996}$.
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