Sequence and Series revision notes

Key Concepts & Definitions

Sequence

An arrangement of numbers in a definite order according to some rule. Theoretically, a sequence can be regarded as a function whose domain is the set of natural numbers (or a subset of it like {1,2,3,...,k}\{1, 2, 3, ..., k\}{1,2,3,...,k}). The terms are denoted as a1,a2,a3,…,ana_1, a_2, a_3, \dots, a_na1​,a2​,a3​,…,an​, where ana_nan​ is the general or nnnth term.

Finite vs Infinite Sequence

A sequence containing a finite number of terms is a finite sequence. If it never ends, it is an infinite sequence.

Progression

Sequences that follow specific, predictable patterns are called progressions.

Fibonacci Sequence

A sequence where there is no single algebraic formula for the nnnth term, but terms are generated by a recurrence relation: a1=a2=1a_1 = a_2 = 1a1​=a2​=1, and an=an−1+an−2a_n = a_{n-1} + a_{n-2}an​=an−1​+an−2​ for n>2n > 2n>2.JEE TIPThe ratio of consecutive Fibonacci numbers converges to the Golden Ratio ϕ≈1.618\phi \approx 1.618ϕ≈1.618.

Series

The sum of the terms of a sequence, expressed as a1+a2+a3+⋯+ana_1 + a_2 + a_3 + \dots + a_na1​+a2​+a3​+⋯+an​. It is abbreviated using the compact sigma (Σ\SigmaΣ) notation: ∑k=1nak\sum_{k=1}^{n} a_k∑k=1n​ak​. A series refers to the indicated sum, not the final evaluated number itself.

Arithmetic Progression (A.P.)

A sequence where the difference between any consecutive term is constant.

• General Term: $a_n = a + (n-1)d$, where $a$ is the first term and $d$ is the common difference.
• Sum of $n$ terms: $S_n = \frac{n}{2}[2a + (n-1)d]$ or $S_n = \frac{n}{2}[a + l]$, where $l$ is the last term.
• Properties:
  • If $a, b, c$ are in A.P., then $2b = a + c$.
  • JEE TIPIf the sum of $n$ terms of a sequence is a quadratic polynomial in $n$ (i.e., $S_n = An^2 + Bn$), the sequence is ALWAYS an A.P., and its common difference is $2A$.
  • JEE TIPIf you need to assume 3 terms in an A.P., choose $a-d, a, a+d$. For 4 terms, choose $a-3d, a-d, a+d, a+3d$ to simplify summation problems.

Geometric Progression (G.P.)

A sequence where every term except the first bears a constant ratio to the term immediately preceding it.

• General Term: For a G.P. with first term $a$ and common ratio $r$, the $n$th term is $a_n = ar^{n-1}$.
• Sum of $n$ terms ($S_n$):
  • If $r = 1$: $S_n = na$.
  • If $r \neq 1$: $S_n = \frac{a(1-r^n)}{1-r}$ or $S_n = \frac{a(r^n-1)}{r-1}$.
• Infinite Geometric Series: If $|r| < 1$, the sum of an infinite G.P. converges to $S_\infty = \frac{a}{1-r}$.
• Properties:
  • If $a, b, c$ are in G.P., then $b^2 = ac$.
  • If $a_1, a_2, \dots, a_n$ is a G.P., then their logarithms $\log(a_1), \log(a_2), \dots$ form an A.P.
  • JEE TIPThe product of the terms equidistant from the beginning and the end of a finite G.P. is constant and equals the product of the first and last terms ($a_k \cdot a_{n-k+1} = a \cdot l$).
  • JEE TIPFor recurring decimal sequences (like $7, 77, 777 \dots$), factor out the digit, multiply and divide by 9, and write the sequence as $(10-1), (10^2-1), \dots$ to convert it into a sum of a G.P. and a constant.

Harmonic Progression (H.P.)

A sequence is an H.P. if the reciprocals of its terms form an Arithmetic Progression.

• General Term: $a_n = \frac{1}{a + (n-1)d}$
• JEE TIPThere is NO general formula for the sum of $n$ terms of an H.P. Questions asking for sum of H.P. terms usually involve telescopic cancellation.
• If $a, b, c$ are in H.P., then $b = \frac{2ac}{a+c}$.

Arithmetico-Geometric Progression (A.G.P.)

A sequence formed by multiplying the corresponding terms of an A.P. and a G.P.

• Standard Form: $a, (a+d)r, (a+2d)r^2, \dots, [a+(n-1)d]r^{n-1}$.
• Sum to $n$ 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 ($|r| < 1$): $S_\infty = \frac{a}{1-r} + \frac{dr}{(1-r)^2}$.
• JEE TIPNever memorize the finite A.G.P. formula. Always use the derivation method: write $S_n$, multiply the entire equation by the common ratio $r$, shift all terms to the right by one position, and subtract to create a pure G.P.

Means and Their Inequalities

• Arithmetic Mean (A.M.): Between $a$ and $b$, $A = \frac{a+b}{2}$. To insert $n$ A.M.s ($A_1, A_2, \dots, A_n$) between $a$ and $b$, use common difference $d = \frac{b-a}{n+1}$.
• Geometric Mean (G.M.): Between $a$ and $b$, $G = \sqrt{ab}$.
  • To insert $n$ G.M.s ($G_1, G_2, \dots, G_n$) between positive numbers $a$ and $b$, the resulting sequence $a, G_1, G_2, \dots, G_n, b$ is a G.P..
  • The common ratio is $r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$. Thus $G_k = a \left(\frac{b}{a}\right)^{\frac{k}{n+1}}$.
  • JEE TIPThe product of $n$ G.M.s inserted between $a$ and $b$ equals the $n$th power of the single G.M. between them: $G_1 \cdot G_2 \dots G_n = (\sqrt{ab})^n$.
• Harmonic Mean (H.M.): Between $a$ and $b$, $H = \frac{2ab}{a+b}$.
• Fundamental Inequality: For any set of positive real numbers, $A.M. \geq G.M. \geq H.M.$
  • Equality holds true only when all the numbers are identical.
  • Proof for two numbers: $A - G = \frac{a+b}{2} - \sqrt{ab} = \frac{(\sqrt{a} - \sqrt{b})^2}{2} \geq 0 \implies A \geq G$.
  • JEE TIPIf A.M. ($A$) and G.M. ($G$) of the roots of a quadratic equation are given, the equation is $x^2 - 2Ax + G^2 = 0$. Roots are given by $A \pm \sqrt{A^2 - G^2}$.

Special Series and Summation Techniques

• 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: $\sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2 = (\sum_{k=1}^n k)^2$.
• Method of Differences: If the differences of successive terms of a sequence form an A.P. or G.P., assume the general term $a_n$ as a polynomial (e.g., $An^2 + Bn + C$) or a combination of polynomial and exponential functions, then equate coefficients.
• Telescopic Sums (V-N Method):JEE TIPBreak the general term $t_n$ into a difference of two consecutive terms of another sequence, i.e., $t_n = V_n - V_{n-1}$ or $t_n = V_{n+1} - V_n$. When you sum $t_n$, all intermediate terms cancel out, leaving only the first and last boundary terms.

Conditions & Limitations

• AM $\geq$ GM $\geq$ HM Inequality: This is STRICTLY APPLICABLE ONLY FOR POSITIVE REAL NUMBERS. Applying it to negative numbers or complex numbers will yield mathematically disastrous results.
• Sum of Infinite G.P.: The formula $S_\infty = \frac{a}{1-r}$ can ONLY be used if the common ratio $r$ satisfies $-1 < r < 1$ (i.e., $|r| < 1$). If $|r| \geq 1$, the series diverges and has no finite sum.
• Common Ratio $r = 1$: The standard G.P. sum formula $\frac{a(1-r^n)}{1-r}$ is undefined for $r=1$ due to division by zero. Use $S_n = na$ instead.
• Geometric Mean defined: The GM of $a$ and $b$ ($\sqrt{ab}$) is only purely real and strictly defined in standard progression contexts when $a$ and $b$ have the same sign. If $a$ and $b$ have opposite signs, their GM is strictly non-real (imaginary), and they cannot form a real sequence.

Previous Year JEE Topics

• Telescopic Series and Cancellation: Creating $V_r - V_{r-1}$ forms using partial fractions or rationalization.
• Optimization using AM $\geq$ GM $\geq$ HM: Finding the minimum value of algebraic or trigonometric expressions by exploiting the AM-GM inequality (often requiring breaking terms into multiple equal parts to adjust powers).
• Properties of A.P., G.P., H.P. Combined: Questions stating "$a, b, c$ are in A.P. and $b, c, d$ are in G.P.", requiring simultaneous substitution of progression properties.
• Arithmetico-Geometric Series (AGP): Evaluating exact limits of infinite AGP or tracking telescoping differences.
• Sum of terms in matrix/determinant: Using properties of progressions to reduce complex determinants to 0.

Top 10 JEE MCQ Traps

• [JEE TIP] Trap 1 - The Infinite G.P. Convergence Constraint:

  • Misconception: The infinite geometric sum formula $S_\infty = \frac{a}{1-r}$ can be applied to any progression as long as a common ratio $r$ is visible.
  • Correct Understanding: The formula is strictly valid if and only if the common ratio satisfies $|r| < 1$. Before applying it to variable ratios (e.g., functions of $x$ like $r = 2\sin x$), you must check the domain or restrict the variable to ensure the series does not diverge. If $|r| \ge 1$, the infinite sum does not exist.

• [JEE TIP] Trap 2 - The Positive Real Reality of AM-GM:

  • Misconception: The Arithmetic Mean-Geometric Mean inequality ($\text{A.M.} \ge \text{G.M.}$) can be used to prove that the expression $x + \frac{1}{x}$ is always greater than or equal to $2$.
  • Correct Understanding: The $\text{A.M.} \ge \text{G.M.}$ inequality applies exclusively to non-negative real numbers. If $x$ is negative ($x < 0$), the terms violate this condition. Instead, for negative values, the inequality flips direction: $x + \frac{1}{x} \le -2$. Always verify the sign of your variables before choosing a minimum value boundary.

• [JEE TIP] Trap 3 - The Mean Insertion Count Expansion:

  • Misconception: Inserting $n$ Arithmetic or Geometric Means between two numbers $a$ and $b$ creates a total sequence that contains exactly $n$ terms.
  • Correct Understanding: The full sequence consists of the two original boundary numbers plus the inserted means ($a, M_1, M_2, \dots, M_n, b$), meaning it contains exactly $n + 2$ terms. Consequently, the final term $b$ must be treated algebraically as the $(n+2)^{\text{th}}$ term of the progression, not the $n^{\text{th}}$ term.

• [JEE TIP] Trap 4 - Variable Residue in AM-GM Optimization:

  • Misconception: To find the minimum value of $x^2 + \frac{2}{x}$ for $x > 0$, you can directly group the two expressions into a two-term mean: $\frac{x^2 + 2/x}{2} \ge \sqrt{2x}$.
  • Correct Understanding: Applying the inequality this way leaves a variable $x$ on the Right-Hand Side (RHS), which fails to provide a constant minimum value. To optimize correctly, you must split the terms so that the variables cancel out completely in the product. Splitting the fraction yields three terms: $x^2 + \frac{1}{x} + \frac{1}{x}$. Applying a three-term $\text{A.M.} \ge \text{G.M.}$ gives: $\frac{x^2 + 1/x + 1/x}{3} \ge \sqrt[3]{x^2 \cdot \frac{1}{x} \cdot \frac{1}{x}} = 1$, which cleanly evaluates to a minimum value of $3$.

• [JEE TIP] Trap 5 - The Sum vs. Term Function Confusion:

  • Misconception: When given a quadratic equation representing the sum of a sequence, such as $S_n = 5n^2 + 3n$, this algebraic expression can be treated directly as the formula for individual terms ($a_n$).
  • Correct Understanding: The function $S_n$ accumulates all terms up to $n$. To extract the specific formula for the $n^{\text{th}}$ individual term, you must compute the difference between successive sums: $a_n = S_n - S_{n-1}$. Confusing the sum function with the term function will completely corrupt your progression equations.

• [JEE TIP] Trap 6 - Even Root Progression Bifurcation:

  • Misconception: Solving a higher-degree ratio equation like $r^4 = 16$ yields a single unique common ratio $r = 2$, which defines a single unique geometric progression.
  • Correct Understanding: Even-powered roots generate both positive and negative real solutions: $r = \pm 2$. This splits the problem into two entirely different valid sequences: a monotonically increasing progression ($r = 2$) and an oscillating, alternating progression ($r = -2$). Failing to account for the negative root will cause you to miss valid answers in multiple-correct questions.

• [JEE TIP] Trap 7 - The Non-Existent Harmonic Sum Formula:

  • Misconception: Because Arithmetic and Geometric progressions have elegant, clean summation formulas, a similar closed-form algebraic formula must exist to compute the sum ($S_n$) of a Harmonic Progression (H.P.).
  • Correct Understanding: There is no closed-form algebraic formula for the sum of an H.P. Attempting to derive one during the exam will waste critical time. If you encounter a summation problem involving harmonic terms, look for alternative structural methods such as telescoping terms ($\sum \frac{1}{n(n+1)}$) or bounding the series using inequalities.

• [JEE TIP] Trap 8 - Determinant Row Progression Reductions:

  • Misconception: Evaluating a $3 \times 3$ matrix determinant where the row elements consist of algebraic linear progressions requires expanding the entire polynomial expression from scratch.
  • Correct Understanding: You can use standard row operations to solve these instantly. If the elements of consecutive rows form arithmetic progressions, applying row operations like $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_2$ reduces the entries to identical rows of constant common differences. This creates proportional rows, instantly collapsing the value of the determinant to exactly $0$.

• [JEE TIP] Trap 9 - The Continuous Derivative Calculus Fallacy:

  • Misconception: To locate the maximum or minimum value of a discrete sequence sum like $S_n = \frac{n(n+1)}{2}$, you can differentiate the expression with respect to $n$ and set the derivative to zero using standard calculus methods.
  • Correct Understanding: Progressions and series are discrete functions whose domains are strictly restricted to the set of natural numbers ($\mathbb{N}$). Because they are not continuous, they cannot be differentiated. To analyze their increasing or decreasing behavior, you must use discrete analysis by evaluating the sign of the difference between consecutive terms: $a_{n+1} - a_n > 0$.

• [JEE TIP] Trap 10 - The Trivial Constant Sequence Monopoly:

  • Misconception: Every arithmetic progression must have a non-zero common difference ($d \neq 0$), and every geometric progression must have a common ratio other than one ($r \neq 1$).
  • Correct Understanding: A uniform constant sequence, such as $2, 2, 2, 2, \dots$, is simultaneously an A.P. ($d = 0$), a G.P. ($r = 1$), and an H.P. JEE multiple-choice questions often include boundary cases where checking a constant sequence reveals that an option is technically true or exposes an edge-case contradiction. Always test the constant sequence option when constraints do not explicitly rule it out.