Algebra/Arithmetic Progression (AP)

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

Example 1:
A frog jumps every 2 seconds. If we start from the very first second the frog jump we take it as ${\displaystyle t=0}$. The next jump will be on ${\displaystyle t=2}$. As the frog jumps every 2 seconds so; ${\displaystyle t=0+2\rightarrow t=2}$

So can you tell when the next jump will be? If you said ${\displaystyle t=4}$ then yes, you are correct. Do you remember how we got ${\displaystyle t=2}$? Now how did you get ${\displaystyle t=4}$? The same way right? Well what you did was you added the difference ${\displaystyle d=2}$ to previous term (which was 2). So to get next term of AP you always add the common difference to current term, and to get previous term you always subtract the common difference from current term. A more general formula to get the ‘n’th term of an AP is:
${\displaystyle a_{n}=a_{1}+(n-1)d}$

Where ${\displaystyle a_{1}}$ is the first term and d is the common difference of the AP. So the general terms of AP are:
${\displaystyle a_{1},\;a_{1}+d,\;a_{1}+2d,\;a_{1}+3d,\;\dots \;a_{1}+(n-1)d}$

Finding the sum of ‘n’ terms of an arithmetic sequence

When needed to find the sum of ‘n’ terms of an AP, you use:
${\displaystyle s_{n}={\frac {n}{2}}(2a_{1}+(n-1)d)}$

!If you have the first and last term you can use a short-cut formula to obtain the sum.
${\displaystyle s_{n}={\frac {n(a_{1}+a_{n})}{2}}}$

Where ${\displaystyle a_{n}}$ is the last term.

Notes

1. The sum of infinite arithmetic sequence is essentially infinite, but if ${\displaystyle a=b=0}$, then the sum is 0.