# 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.