Progression Formulas

A succession of numbers formed and arranged in a definite order according to certain definite rule, is called a progression.
We have two type of progressions
1. Arithmetic Progression (A.P.):
If each term of a progression differs from its preceding term by a constant, then such a progression is called an arithmetic progression. The constant is called as common difference of the Arithmetic Progression.
For example:
1, 3, 5, 7, 9...
this sequence is in Arithmetic Progression by common difference 2.
There are few important results for Arithmetic Progression, which you should be aware of.

Nth term in Arithmetic Progression(A.P.)= a + (n-1)d

Sum of n terms of this A.P.
\begin{aligned}
= \frac{n}{2}[2a+(n-1)d] \\
\text{or } \frac{n}{2}(\text{first term + last term})
\end{aligned}

Other Important Results:
\begin{aligned}
(1+2+3+..+n) = \frac{n(n+1)}{2} \\
(1^2+2^2+3^2+..+n^2) = \frac{n(n+1)(2n+1)}{6}\\
(1^3+2^3+3^3+..+n^3) = \frac{n^2(n+1)^2}{4}\\

\end{aligned}

2. Geometrical Progression (G.P.):
A progression of numbers in which every term bears a constant ratio with its preceding term, is called a Geometrical Progression. The constant ratio is called the common ratio of the G.P.
Example:
\begin{aligned}
a, ar,ar^2, ar^3,...
\end{aligned}

In Geometrical Progression(G.P.) nth term is
\begin{aligned}
= ar^{n-1}
\end{aligned}

In Geometrical Progression(G.P.) Sum of the n terms
\begin{aligned}
= \frac{a(1-r^n)}{1-r}
\end{aligned}