# JEE Sequence

This page will teach you how to master JEE Sequences. 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 sequences are arithmetic progression, geometric progression, harmonic progression, AM-GM-HM inequality and its applications.

### Arithmetic Progression (AP)

- General term: \(t_n=a+(n-1)d\)
- Sum of the terms: \(S_n=\frac n2 \left( 2a+(n-1)d \right) \)
- Arithmetic mean and its property: \[(\text{AM of any } n \text{ numbers } a_1,a_2,\ldots,a_n)=\frac{a_1+a_2+\cdots+a_n}{n}\]
- Insertion of arithmetic means between two numbers:

If \(a\) and \(b\) are two given numbers and \(a,A_1,A_2,\ldots,A_n,b\) are in AP, then \(A_1,A_2,\ldots,A_n\) are \(n\) AM's between \(a\) and \(b:\) \[\begin{array} &A_1=a+\frac{b-a}{n+1}, &A_2=a+\frac{2(b-a)}{n+1}, &\ldots, &A_n=a+\frac{n(b-a)}{n+1}.\end{array}\]

### Geometric Progression (GP)

- General term: \(t_n=ar^{n-1}\)
- Sum of the terms: \(S_n-\dfrac{a(r^n-1)}{r-1}\)
- Geometric mean and its property: \[(\text{GM of any } n \text{ numbers } a_1,a_2,\ldots,a_n)=\left( a_1 \cdot a_2 \cdot \cdot \cdot a_n \right)^{\frac 1n}\]
- Insertion of geometric means between two numbers:

If \(a\) and \(b\) are two given numbers and \(a, G_1, G_2, \ldots, G_n, b\) are in GP, then \(G_1,G_2,\ldots,G_n\) are \(n\) GM's between \(a\) and \(b:\) \[\begin{array} &G_1=a\left( \frac ba \right)^{\frac{1}{n+1}}, &G_2=a\left( \frac ba \right)^{\frac{2}{n+1}}, &\ldots, &G_n=a\left( \frac ba \right)^{\frac{n}{n+1}}. \end{array}\]

### Harmonic Progression (H.P.)

- General term: \(\dfrac{1}{a_n}=\dfrac{1}{a}+(n-1)d\)
- Harmonic mean and its property:

If HM of any \(n\) numbers \(a_1,a_2,\ldots,a_n\) is \(H\), then \[H=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}.\] - Insertion of harmonic means between two numbers:

If \(a\) and \(b\) are two given numbers and \(a,H_1,H_2,\ldots,H_n,b\) are in HP, then \(H_1,H_2,\ldots,H_n\) are \(n\) HM's between \(a\) and \(b:\) \[\begin{array} &\frac{1}{H_1}=\frac{1}{a}+\frac{\left( \frac 1b - \frac 1a \right)}{n+1}, &\frac{1}{H_2}=\frac{1}{a}+\frac{ 2 \left( \frac 1b - \frac 1a \right)}{n+1}, &\ldots, &\frac{1}{H_n}=\frac{1}{a}+\frac{n \left( \frac 1b - \frac 1a \right)}{n+1}. \end{array}\]

### AM-GM-HM inequality and its applications

- If \(a_1,a_2,\ldots,a_n\) are all positive real numbers, then

\[\dfrac{a_1+a_2+...+a_n}{n} \geq \left( a_1 \cdot a_2 \cdot \cdot \cdot a_n \right)^{\frac 1n} \geq \dfrac{n}{\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}}.\]

## 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 Sequences, move on to JEE Series.