Question: I'm trying to work through example 2, from [here](http://torch.ch/docs/five-simple-examples.html#_).

We start by defining a symmetric positive definite matrix $A$:

$\begin{pmatrix} 
1.2054 & 0.6593 &1.2299 & 1.2577 & 1.0083\\
 0.6593 & 0.5179 & 0.5460 & 0.8562 & 0.4608\\
 1.2299 & 0.5460 & 2.6713 & 1.8403 & 1.6162\\
 1.2577 & 0.8562 & 1.8403 & 2.5114 & 1.6419\\
 1.0083 & 0.4608 & 1.6162 & 1.6419 & 1.4266
\end{pmatrix}$.

Next, we define a vector $b$ with random numbers as entries:

$\begin{pmatrix}
 0.0258\\
 0.1957\\
 0.9065\\
 0.3823\\
 0.7864
\end{pmatrix}$

Then, we compute $xs=A^{-1}b$. 

Finally, we feed this the quadratic form:

$0.5xs^T Axs^T-b^Txs$

and, with the my numbers, get $-4.46333$, which doesn't appear anywhere in the matrix $A$. So if this is supposed to be an exact minimum, what's it supposed to be an exact minimum of?

Answer: 
$$10^{2(n+1)}+(-1)^{(n+1)+1}=100\cdot10^{2n}-(-1)^{n+1}=101\cdot10^{2n}-\underbrace{(10^{2n}+(-1)^{n+1})}\_{\text{multiple of }101}$$
Notice the following steps of mathematical induction, 

1. Setting $n=1$ in the given number $10^{2n}+(-1)^{n+1}$ gives $$10^{2\cdot 1}+(-1)^{1+1}=101$$ above number is divisible by $101$ for $n=1$
2. assume that $10^{2n}+(-1)^{n+1}$ is divisible by $101$ for $n=k$ then $$10^{2k}+(-1)^{k+1}=101m$$
or $$10^{2k}=101m-(-1)^{k+1}\tag 1$$
where, $m$ is some integer
3. setting $n=k+1$, $$10^{2(k+1)}+(-1)^{k+1+1}$$
$$=100\cdot 10^{2k}-(-1)^{k+1}$$
setting the value of $10^{2k}$ from (1), 
$$=100(101m-(-1)^{k+1})-(-1)^{k+1}$$
$$=101(100m)-101(-1)^{k+1}$$
$$=101(100m-(-1)^{k+1})$$

since, $(100m-(-1)^{k+1})$ is an integer hence the number $101(100m-(-1)^{k+1})$ is divisible by $101$ 

thus $10^{2n}+(-1)^{n+1}$ is divisible by $101$ for $n=k+1$

hence, $10^{2n}+(-1)^{n+1}$ is divisible by $101$ for all integers $n\ge 1$