Sequences & Series Cheat Sheet for JEE Main

This cheat sheet provides a concise yet comprehensive overview of Sequences and Series, essential for JEE Main preparation. We'll cover key definitions, formulas, and problem-solving techniques to help you master this high-scoring topic.

Key Concepts

Sequence: An ordered list of numbers, also known as a progression. Each number in the sequence is called a term. A sequence can be finite or infinite. Series: The sum of the terms of a sequence. Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. This constant difference is called the common difference ($d$). Geometric Progression (GP): A sequence where the ratio of consecutive terms is constant. This constant ratio is called the common ratio ($r$). Harmonic Progression (HP): A sequence whose reciprocals are in Arithmetic Progression. There is no direct formula for the sum of an HP. Arithmetic Mean (AM): For two numbers $a$ and $b$, the AM is $\frac{a+b}{2}$. For $n$ numbers $a_1, a_2, ..., a_n$, the AM is $\frac{a_1+a_2+...+a_n}{n}$. Geometric Mean (GM): For two positive numbers $a$ and $b$, the GM is $\sqrt{ab}$. For $n$ positive numbers $a_1, a_2, ..., a_n$, the GM is $\sqrt[n]{a_1 a_2 ... a_n}$. Harmonic Mean (HM): For two positive numbers $a$ and $b$, the HM is $\frac{2}{\frac{1}{a} + \frac{1}{b}} = \frac{2ab}{a+b}$. For $n$ positive numbers $a_1, a_2, ..., a_n$, the HM is $\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}}$. Arithmetic-Geometric Progression (AGP): A series formed by multiplying the corresponding terms of an AP and a GP. Sum to infinity of a GP: Exists only when the common ratio $|r| < 1$. $\sum n, \sum n^2, \sum n^3$: Standard sums of powers of natural numbers.

Formulas & Results

1. Arithmetic Progression (AP):

• General term ($n^{th}$ term): $a_n = a + (n-1)d$
• Sum of first $n$ terms: $S_n = \frac{n}{2}(a + a_n) = \frac{n}{2}(2a + (n-1)d)$
• If $a, b, c$ are in AP, then $2b = a+c$.
• Arithmetic Mean between $a$ and $b$: $A = \frac{a+b}{2}$
• $n$ arithmetic means between $a$ and $b$: $A_k = a + k\left(\frac{b-a}{n+1}\right)$ for $k=1, 2, ..., n$.

2. Geometric Progression (GP):

• General term ($n^{th}$ term): $a_n = ar^{n-1}$
• Sum of first $n$ terms: $S_n = \frac{a(r^n - 1)}{r-1}$ for $r \neq 1$.
• If $a, b, c$ are in GP, then $b^2 = ac$.
• Sum of an infinite GP (when $|r|<1$): $S_\infty = \frac{a}{1-r}$
• Geometric Mean between $a$ and $b$ (positive $a, b$): $G = \sqrt{ab}$
• $n$ geometric means between $a$ and $b$: $G_k = a\left(\frac{b}{a}\right)^{\frac{k}{n+1}}$ for $k=1, 2, ..., n$.

3. Harmonic Progression (HP):

• If $a_1, a_2, ..., a_n$ are in HP, then $\frac{1}{a_1}, \frac{1}{a_2}, ..., \frac{1}{a_n}$ are in AP.
• The $n^{th}$ term of an HP whose reciprocals are in AP with first term $A$ and common difference $D$ is given by $h_n = \frac{1}{A + (n-1)D}$.
• Harmonic Mean between $a$ and $b$: $H = \frac{2ab}{a+b}$

4. Relations between AM, GM, HM:

• For two positive numbers $a$ and $b$: $G^2 = AH$.
• For any set of positive numbers: $AM \ge GM \ge HM$. Equality holds if all numbers are equal.

5. Sums of Special Series:

• Sum of first $n$ natural numbers: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$
• Sum of squares of first $n$ natural numbers: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$
• Sum of cubes of first $n$ natural numbers: $\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2$

6. Arithmetic-Geometric Progression (AGP): A typical AGP is of the form $a, (a+d)r, (a+2d)r^2, ..., (a+(n-1)d)r^{n-1}$. Sum of $n$ terms: $S_n = \frac{a}{1-r} + \frac{dr(1-r^{n-1})}{(1-r)^2} - \frac{(a+(n-1)d)r^n}{1-r}$ Sum to infinity (when $|r|<1$): $S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}$

Worked Examples

Example 1: Arithmetic Progression Find the sum of all two-digit numbers which are divisible by 3.

Solution: The two-digit numbers divisible by 3 form an AP: $12, 15, 18, ..., 99$. Here, the first term $a = 12$. The common difference $d = 15 - 12 = 3$. The last term $a_n = 99$. We need to find the number of terms, $n$. Using $a_n = a + (n-1)d$: $99 = 12 + (n-1)3$ $87 = (n-1)3$ $29 = n-1$ $n = 30$ Now, find the sum of these $n$ terms using $S_n = \frac{n}{2}(a + a_n)$: $S_{30} = \frac{30}{2}(12 + 99)$ $S_{30} = 15(111)$ $S_{30} = 1665$ Thus, the sum of all two-digit numbers divisible by 3 is 1665.

Example 2: Geometric Progression If the product of three consecutive terms of a GP is 216 and the sum of their products taken in pairs is 156, find the terms.

Solution: Let the three consecutive terms of the GP be $\frac{a}{r}, a, ar$. Given, product of terms = 216: $\left(\frac{a}{r}\right)(a)(ar) = 216$ $a^3 = 216$ $a = 6$ Given, sum of their products taken in pairs = 156: $\left(\frac{a}{r}\right)(a) + (a)(ar) + \left(\frac{a}{r}\right)(ar) = 156$ $\frac{a^2}{r} + a^2r + a^2 = 156$ Substitute $a=6$: $\frac{36}{r} + 36r + 36 = 156$ $\frac{36}{r} + 36r = 120$ Divide by 12: $\frac{3}{r} + 3r = 10$ $3 + 3r^2 = 10r$ $3r^2 - 10r + 3 = 0$ $(3r-1)(r-3) = 0$ So, $r = \frac{1}{3}$ or $r = 3$.

If $a=6, r=3$, the terms are $\frac{6}{3}, 6, 6 \times 3 \Rightarrow 2, 6, 18$. If $a=6, r=\frac{1}{3}$, the terms are $\frac{6}{1/3}, 6, 6 \times \frac{1}{3} \Rightarrow 18, 6, 2$. In both cases, the terms are $2, 6, 18$.

Example 3: Sum of AGP Find the sum to infinity of the series $1 + \frac{2}{3} + \frac{3}{3^2} + \frac{4}{3^3} + \dots$

Solution: This is an Arithmetic-Geometric Progression (AGP). Let $S = 1 + \frac{2}{3} + \frac{3}{3^2} + \frac{4}{3^3} + \dots \quad \dots(1)$ Here, the AP is $1, 2, 3, 4, \dots$ ($a=1, d=1$) The GP is $1, \frac{1}{3}, \frac{1}{3^2}, \frac{1}{3^3}, \dots$ ($a_{GP}=1, r=\frac{1}{3}$) Multiply (1) by the common ratio of GP, $r = \frac{1}{3}$: $\frac{1}{3}S = \frac{1}{3} + \frac{2}{3^2} + \frac{3}{3^3} + \frac{4}{3^4} + \dots \quad \dots(2)$ Subtract (2) from (1): $S - \frac{1}{3}S = (1 + \frac{2}{3} + \frac{3}{3^2} + \dots) - (\frac{1}{3} + \frac{2}{3^2} + \frac{3}{3^3} + \dots)$ $\frac{2}{3}S = 1 + (\frac{2}{3} - \frac{1}{3}) + (\frac{3}{3^2} - \frac{2}{3^2}) + (\frac{4}{3^3} - \frac{3}{3^3}) + \dots$ $\frac{2}{3}S = 1 + \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \dots$ The RHS is an infinite GP with first term $A=1$ and common ratio $R=\frac{1}{3}$. Since $|R|<1$, the sum of this infinite GP is $S_{\infty, GP} = \frac{A}{1-R}$. $S_{\infty, GP} = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = \frac{3}{2}$. So, $\frac{2}{3}S = \frac{3}{2}$ $S = \frac{3}{2} \times \frac{3}{2} = \frac{9}{4}$.

Common Mistakes

• Confusing AP, GP, HP: Always clearly identify the type of progression before applying formulas. For HP, convert to AP and work with reciprocals.
• Incorrect common ratio/difference: Double-check calculations for $d$ and $r$. For GP, $r = \frac{a_k}{a_{k-1}}$ not $\frac{a_{k-1}}{a_k}$.
• Applying sum to infinity formula incorrectly: $S_\infty = \frac{a}{1-r}$ is valid ONLY when $|r|<1$.
• Error in $n^{th}$ term for AP/GP: Remember $a_n = a + (n-1)d$ and $a_n = ar^{n-1}$. The exponent for $r$ is $n-1$, not $n$.
• Forgetting conditions for AM-GM-HM inequality: The inequality $AM \ge GM \ge HM$ holds only for positive numbers.
• Algebraic errors in solving equations: Be careful while solving for $a, d, r, n$ from given conditions.
• Not checking for $r=1$ in GP sum: The formula for $S_n$ for GP changes when $r=1$. In such cases, $S_n = na$.

Quick Revision Checklist

• Can you define AP, GP, HP and identify their common properties ($d, r$)?
• Do you know the formulas for $n^{th}$ term of AP and GP?
• Can you compute the sum of $n$ terms for AP and GP?
• Are you familiar with the sum to infinity formula for GP and its condition?
• Can you apply relationship between AM, GM, HM? ($G^2=AH$, $AM \ge GM \ge HM$)
• Are you able to solve problems involving selecting terms in AP/GP? (e.g., $a-d, a, a+d$ for 3 terms in AP)
• Do you know how to find the general term and sum of an AGP?
• Can you recall standard sum formulas: $\sum n, \sum n^2, \sum n^3$?
• Are you comfortable with the method of differences for arbitrary series?
• Can you handle problems where terms of sequence are related to $\log$ or $\exp$?