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.

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 $b:$ $\begin{array}{c}&A_1=a+\frac{b-a}{n+1}, &A_2=a+\frac{2(b-a)}{n+1}, &\ldots, &A_n=a+\frac{n(b-a)}{n+1}.\end{array}$

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 $b:$ $\begin{array}{c}&G_1=a\left( \frac ba \right)^{\frac{1}{n+1}}, &G_2=a\left( \frac ba \right)^{\frac{2}{n+1}}, &\ldots, &G_n=a\left( \frac ba \right)^{\frac{n}{n+1}}. \end{array}$

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 $b:$ $\begin{array}{c}&\frac{1}{H_1}=\frac{1}{a}+\frac{\left( \frac 1b - \frac 1a \right)}{n+1}, &\frac{1}{H_2}=\frac{1}{a}+\frac{ 2 \left( \frac 1b - \frac 1a \right)}{n+1}, &\ldots, &\frac{1}{H_n}=\frac{1}{a}+\frac{n \left( \frac 1b - \frac 1a \right)}{n+1}. \end{array}$

AM-GM-HM inequality and its applications

• If $a_1,a_2,\ldots,a_n$ are all positive real numbers, then

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

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

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

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

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

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Once you are confident of JEE Sequences, move on to JEE Series.