Understanding Sequence and Series in Mathematics

A sequence is an ordered set of numbers following a specific pattern, while a series is the sum of the terms of a sequence. Both concepts admit rigorous algebraic formulation and are foundational for the study of mathematical progressions and summation techniques in JEE syllabus.

Notation and Formal Description of Sequences and Series

A sequence is denoted as $\{a_n\}$, where $a_n$ represents the $n$-th term, and $n \in \mathbb{N}$. A series is the formal sum $\sum_{n=1}^{\infty} a_n$, representing either a finite or an infinite aggregation of sequence terms.

A sequence is finite if it contains a specific number of terms and infinite if no last term is specified. A series is finite or infinite depending upon the length of its underlying sequence.

General Term (nth Term) Formulations in Progressions

For an arithmetic progression (AP) with initial term $a$ and common difference $d$, the general term is $a_n = a + (n-1)d$. For a geometric progression (GP) with first term $a$ and common ratio $r$, the $n$-th term is $a_n = a r^{n-1}$. Harmonic progression (HP) terms are defined as reciprocals of an AP: $a_n = \dfrac{1}{a + (n-1)d}$.

Summation Expressions for Standard Progressions

The sum of the first $n$ terms in an arithmetic progression is $S_n = \dfrac{n}{2}[2a + (n-1)d]$. The sum in a geometric progression is $S_n = \dfrac{a(1 - r^n)}{1 - r}$, where $r \neq 1$. For $|r| < 1$, the sum to infinity is $S_\infty = \dfrac{a}{1 - r}$.

Sums of the first $n$ natural numbers, their squares, and cubes are respectively given by $S_1 = \dfrac{n(n+1)}{2}$, $S_2 = \dfrac{n(n+1)(2n+1)}{6}$, and $S_3 = \left[\dfrac{n(n+1)}{2}\right]^2$.

Arithmetic Mean, Geometric Mean, and Harmonic Mean in Progressions

Given two numbers $a$ and $b$, their arithmetic mean is $A = \dfrac{a+b}{2}$, their geometric mean is $G = \sqrt{ab}$, and their harmonic mean is $H = \dfrac{2ab}{a+b}$. Inserting $n$ arithmetic means between $a$ and $b$ forms an AP, while inserting $n$ geometric means requires a GP structure.

Conceptual relationships: In a set of positive real numbers, $A \geq G \geq H$.

Fibonacci Sequence and Recurrence Relations

The Fibonacci sequence is defined recursively by $F_0 = 0$, $F_1 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Such recurrence structures generalize to linear recurrences of the form $a_n = p a_{n-1} + q a_{n-2}$.

Algebraic Properties of Arithmetic and Geometric Progressions

An arithmetic progression has equal differences $a_{n} - a_{n-1} = d$ for all $n \geq 2$. A geometric progression satisfies $a_n / a_{n-1} = r$ for $n \geq 2$. In a harmonic progression, reciprocal terms constitute an AP.

For sequence transformation, direct addition, subtraction, multiplication, or division by constants preserves arithmetic or geometric structures under certain conditions.

Evaluation of Series Involving Special Patterns

For series where the $n$-th term is given by a general formula, summation can be carried out by expressing $S_n = \sum_{k=1}^n a_k$ and applying methods such as termwise summation, telescoping, or use of mathematical induction.

Utilize algebraic manipulation to handle terms with alternating signs, squares, or cubes, referencing known formulae as required.

Solution Techniques for Typical JEE Problems on Sequences and Series

• Finding the general term
• Determining the sum of terms
• Identifying type of progression
• Discovering inserted means
• Solving equations involving series
• Analyzing recurrence relations

Practice is available from Sequences And Series Practice Paper and related resources to reinforce computation and exam strategies.

Illustrative Worked Examples in Sequences and Series

Example. Determine the 21st term and sum of the first 21 terms for the sequence: $4, 7, 10, \ldots$

Given $a = 4$, $d = 3$.

The $n$-th term is $a_n = 4 + (n - 1) \times 3 = 3n + 1$.

21st term: $a_{21} = 3 \times 21 + 1 = 64$.

Sum: $S_{21} = \dfrac{21}{2}[2 \times 4 + (21-1) \times 3] = \dfrac{21}{2}[8 + 60] = 21 \times 34 = 714$.

Example. Find the 9th term of the geometric sequence $1, 4, 16, 64, \ldots$

Given $a = 1$, $r = 4$.

$a_9 = 1 \times 4^{8} = 65536$.

Example. The $p$-th term of an AP is $q$ and the $q$-th term is $p$. Find the $r$-th term.

Let $a$ be the first term and $d$ the common difference. Then $a + (p-1)d = q$, $a + (q-1)d = p$.

Subtracting, $(p-q)d = q-p \implies d = -1$. Substituting, $a = p + q - 1$.

$r$-th term: $a + (r-1)d = (p+q-1) + (r-1)(-1) = p + q - r$.

Example. If $x, 1, z$ are in AP and $x, 2, z$ in GP, prove $x, 4, z$ are in HP.

As $x, 1, z$ in AP, $x + z = 2$. As $x, 2, z$ in GP, $z = 4/x$.

Solving $x + 4/x = 2 \implies x^2 - 2x + 4 = 0$.

Now $x, 4, z$ are in HP if $4 = 2xz/(x+z)$. Substituting $z$ and $x+z = 2$ gives $4 = 2x(4/x)/2 = 4$.

For more comprehensive revision, consult Sequences and Series Revision Notes.

Exam Cautions and Structural Distinctions between Sequences and Series

A common error is failing to distinguish between positional arrangement (sequence) and summation (series), especially in series convergence assessment.

Additional advanced practice can be found at Sequences and Series Important Questions and through Understanding Sequence And Series for extended conceptual coverage.