# How many Arithmetic Progressions (having 3 terms) can be made from integers 1 to n?

How many Arithmetic Progressions (having 3 terms) can be made from integers 1 to n? (The numbers in the AP are distinct)

For example if n=6 then number of APs possible are 6

1,2,3
2,3,4
3,4,5
4,5,6
1,3,5
2,4,6

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Note, you don't specifically say that the three numbers are distinct. $1,1,1$ is an arithmetic progression... – Thomas Andrews Dec 1 '12 at 18:56
Yeah I missed that. The numbers are essentially distinct in this question. – Priyam Dec 2 '12 at 18:32
What about 5,3,1? – Hurkyl Dec 2 '12 at 18:35
@Hurkyl it is valid. We will just multiply the answer by 2 as only distinct numbers are allowed – Priyam Dec 19 '12 at 18:17

One can select any tow distinct elements of the same parity. Together with their mean, they make such a progression.

There are $\left\lfloor \frac n2\right\rfloor$ even numbers $\le n$ and $\left\lfloor \frac{n+1}2\right\rfloor$ odd numbers. Since we need to select unordered pairs (or ordered? But you seem not to distinguish between 1,2,3 and 3,2,1), the total number is $$f(n) = {\left\lfloor \frac n2\right\rfloor\choose 2}+{\left\lfloor \frac {n+1}2\right\rfloor\choose 2}.$$ If $n=2m$ is even, this amounts to $$f(n) = 2{m\choose 2}=m(m-1)$$ and if $n=2m+1$ is odd, to $$f(n) = {m\choose 2}+{m+1\choose 2} = m^2.$$ Both may also be summarized as $$f(n) = \left\lfloor\frac n2\right\rfloor\cdot\left\lfloor\frac{n-1}2\right\rfloor.$$

Example: Letting $n=6$, we obtain $f(n)=\left\lfloor\frac 62\right\rfloor\cdot\left\lfloor\frac52\right\rfloor=3\cdot 2=6$.

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Let the three term AP be $a,a+d,a+2d$ where $a,d$ are natural numbers.

So, $a\ge1$ and $a+2d\le n\implies 1\le a\le n-2d \implies d\le \frac{n-1}2$

If $d=1,a$ can assume $1,2,\cdots, n-2$ i.e., $n-2$ values.

If $d=2,a$ can assume $1,2,\cdots, n-4$ i.e., $n-4$ values.

If $n$ is even, $d_{max}=\frac{n-2}2$

for $d=\frac {n-2}2,1\le a\le 2$ i.e., $a$ has 2 values.

If $n$ is even, the number of AP is $(n-2)+(n-4)+\cdots+2=\frac{(n-2)}4(2+n-2)=\frac{n(n-2)}4$

If $n$ is odd, $d_{max}=\frac{n-1}2$

So, in that case the number of AP will be $1+3+\cdots+(n-4)+(n-2)=\frac{(n-1)^2}4$

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