Creating sequences from natural numbers How much different arithmetic sequences can you make from the numbers 1 to 51 ?
Note:
The sequence length has to be 3 numbers.
The difference between each 2 numbers is positive.

There are several ways to count in an organized manner.. We describe three of them.
(Our favourite way is the first, and relatives.)
First way: We need to choose two numbers $a$ and $b$ to serve as the ends of our sequence. The numbers $a$ and $b$ determine a three-term increasing arithmetic sequence  precisely if $\frac{a+b}{2}$ is an integer. So we want $a$ and $b$ to have the same parity.
There are $25$ even numbers in our interval. We can choose $2$ of them in $\binom{25}{2}$ ways. There are $26$ odd numbers in the interval. We can choose $2$ of them in $\binom{26}{2}$ ways. 
Thus the total is $\binom{25}{2}+\binom{26}{2}$.
Second way: There is one three-term increasing arithmetic sequence with "middle" number $2$, there are $2$ with middle number $3$, there are $3$ with middle number $4$,  and so on up to $24$ with middle number $25$.
There are $25$ with middle number $26$.
And by symmetry there are just as many with middle number $\gt 26$ as there are with middle number $\lt 26$.
Thus our count is $2(1/2)(24)(25)+25$, that is, $25^2$. 
Third way: There is $1$ (three-term increasing) arithmetic sequence with common difference $25$, there are $3$ with common difference $24$, there are $5$ with common difference $23$, and so on up to $49$ with common difference $1$.
The sum $1+3+5+\cdots+49$ of the odd numbers up to $49$, is, by a standard formula, equal to $25^2$.
