# JEE Series

This page will teach you how to master JEE Series. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.

## JEE Conceptual Theory

As per JEE syllabus, the main concepts under series are $\sum n , \sum n^2 \text{ and } \sum n^3$ series, arithmetic-geometric series, telescopic sum, telescopic product and successive difference.

### $\sum n , \sum n^2 \text{ and } \sum n^3$ series

Sum of first $n$ natural numbers, abbreviated as $\sum n$, is given by $\sum n=1+2+3+...+n=\frac{n(n+1)}{2}$

Sum of squares of first $n$ natural numbers, abbreviated as $\sum n^2$, is given by $\sum n^2=1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$

Sum of cubes of first $n$ natural numbers, abbreviated as $\sum n^3$, is given by $\sum n^3=1^3+2^3+3^3+...+n^3=\left( \frac{ n(n+1)}{2} \right)^2$

### Arithmetic-Geometric Progression (A.G.P.)

Sum of first $n$ terms of A.G.P. is given by $a+(a+d)r+(a+2d)r^2+(a+3d)r^3+...+\left( a+(n-1)d \right) r^{n-1}$ $=\frac{a}{1-r} +\frac{dr}{(1-r)^2} -\frac{dr^n}{(1-r)^2}- \frac{\left( a+(n-1)d \right) r^n}{1-r}$

Sum of infinite terms of A.G.P. is given by $S_{\infty} =\frac{a}{1-r} +\frac{dr}{(1-r)^2}$

### Telescopic sum and product

Telescopic sum also known as

**method of difference**is used when the general term of the series can be expressed as the difference of two terms by clever manipulation, as $t_r=f(r)-f(r \pm 1)$Telescopic product

### Successive difference

- Finding the $n$th term using "Successive difference" of second order A.P.

## JEE Mains Problems

$\begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array}$

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$\begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array}$

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## JEE Advanced Problems

$\begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array}$

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Once you are confident of JEE Series, move on to JEE Matrices and Determinants.