Introduction to Sequences and Series - (Maths) for JEE Main & Advanced

Sequences

Sequence - A sequence is an ordered list of numbers arranged according to a rule. Each number in the list is called a term of the sequence. A sequence is usually written as (a₁, a₂, a₃, ...) or {a_n} where a_n denotes the nth term.

Example of a simple sequence: (1, 3, 5, 7, 9, 11, ...). This sequence consists of the odd natural numbers and continues indefinitely. If a sequence has infinitely many terms it is called an infinite sequence. If it stops at a last term, for example (1, 3, 5, 7, ..., 131), it is a finite sequence.

Notation and general term

We denote the n^th term by a_n. A sequence is fully specified when a rule to compute a_n for every positive integer n is given, together with the index at which the sequence starts (commonly n = 1).

Fibonacci sequence

The Fibonacci sequence is a well-known example defined by fixed first two terms and a recurrence relation. The first two terms are

a₁ = 1, a₂ = 1.

Every term from the third onwards is the sum of the two preceding terms. Thus

a_n = a_n-1 + a_n-2, for n ≥ 3.

The sequence begins as (1, 1, 2, 3, 5, 8, 13, ...).

An explicit closed form (Binet's formula) for the n^th Fibonacci number is

F_n = (φⁿ - ψⁿ) / √5, where φ = (1 + √5) / 2 and ψ = (1 - √5) / 2.

Other important properties of sequences

• Monotonicity: a sequence is increasing if a_n+1 ≥ a_n for all n, and decreasing if a_n+1 ≤ a_n for all n.
• Boundedness: a sequence is bounded above if there exists M such that a_n ≤ M for all n, and bounded below if there exists m such that a_n ≥ m for all n. If both hold, the sequence is bounded.
• Limit and convergence: a sequence {a_n} has a limit L if the terms approach L as n → ∞; in that case the sequence is said to converge to L. If no finite limit exists, it diverges.

Types of Sequences

• Arithmetic sequence (Arithmetic Progression, AP): The difference between consecutive terms is constant. This constant is called the common difference d. If the first term is a₁, then the n^thterm is

  a_n = a₁ + (n - 1)d.

• Geometric sequence (Geometric Progression, GP): Each term after the first is obtained by multiplying the previous term by a fixed non-zero number r, the common ratio. If a₁is the first term, then

  a_n = a₁ r^n-1.

• Quadratic sequence: A sequence in which the second differencebetween consecutive terms is constant. Such a sequence can be represented by a quadratic polynomial in n:

  a_n = An² + Bn + C, and the constant second difference equals 2A.

• Other types: There are many other sequences such as harmonic sequences (terms are reciprocals of an arithmetic sequence), exponential, factorial, and recurrence-defined sequences (like Fibonacci).

Series

A series is the sum of the terms of a sequence. If {a_n} is a sequence, the sum of the first n terms is the n^th partial sum and is denoted by S_n:

S_n = a₁ + a₂ + ... + a_n = ∑_k=1ⁿ a_k.

An infinite series is the limit of the partial sums as n → ∞. If the limit exists (finite), the series converges; otherwise it diverges.

Important series and formulae

• Sum of first n natural numbers:

  1 + 2 + ... + n = n(n + 1) / 2.

• Sum of first n squares:

  1² + 2² + ... + n² = n(n + 1)(2n + 1) / 6.

• Sum of first n cubes:

  1³ + 2³ + ... + n³ = [n(n + 1) / 2]².

• Sum of an AP (first n terms):

  S_n = n/2 [2a₁ + (n - 1)d] = n/2 [a₁ + a_n].

• Sum of a finite GP (r ≠ 1):

  S_n = a₁ (1 - rⁿ) / (1 - r).

• Sum of an infinite GP (|r| < 1):

  S = a₁ / (1 - r).

• Harmonic series: 1 + 1/2 + 1/3 + 1/4 + ... is called the harmonic series and it diverges (its partial sums grow without bound).

Convergence tests (brief)

• Geometric test: An infinite geometric series with |r| < 1 converges to a₁ / (1 - r); if |r| ≥ 1 it diverges.
• Comparison idea: Sometimes series are compared to known convergent or divergent series to decide convergence.
• For an infinite series ∑ a_n to converge, a necessary condition is that a_n → 0 as n → ∞. If a_n does not tend to 0, the series diverges.

Worked examples

Question: Identify the sequence of the following function n (n+1)

4, 10, 18, 28...

4, 12, 18, 28...

2, 10, 18, 28...

4, 10, 18, 28...

Solution: 
We interpret the question as asking for the sequence produced by the function n(n + 1) when n = 1, 2, 3, ...
Compute values of n(n + 1):
For n = 1, n(n + 1) = 1 × 2 = 2.
For n = 2, n(n + 1) = 2 × 3 = 6.
For n = 3, n(n + 1) = 3 × 4 = 12.
For n = 4, n(n + 1) = 4 × 5 = 20.
Thus the sequence generated by n(n + 1) is (2, 6, 12, 20, ...). None of the option rows as written exactly match this sequence. If instead the intent was n(n + 3), then:
For n = 1, n(n + 3) = 1 × 4 = 4.
For n = 2, n(n + 3) = 2 × 5 = 10.
For n = 3, n(n + 3) = 3 × 6 = 18.
For n = 4, n(n + 3) = 4 × 7 = 28.
So the list (4, 10, 18, 28, ...) corresponds to the function n(n + 3), not to n(n + 1).

Question: Adding first 100 terms in a sequence is

A. term
B. series
C. constant
D. sequence

Solution: 
Adding (summing) the first 100 terms of a sequence produces the 100^th partial sum, which is an example of a series (specifically the sum of the first 100 terms). Therefore the correct choice is B. series.

Additional examples for practise

Example 1: Find the n^th term and the sum of the first n terms of the AP 5, 8, 11, 14, ...

First term a₁ = 5 and common difference d = 3.

The n^th term is

a_n = 5 + (n - 1) × 3 = 3n + 2.

The sum of the first n terms is

S_n = n/2 [2 × 5 + (n - 1) × 3] = n/2 (10 + 3n - 3) = n/2 (3n + 7).

Example 2: For the GP 2, 6, 18, 54, ... find the sum of the first 5 terms.

First term a₁ = 2, common ratio r = 3. Sum of first 5 terms is

S₅ = 2 (1 - 3⁵) / (1 - 3) = 2 (1 - 243) / (-2) = 242.

Applications and remarks

• Sequences and series appear in many applications: calculating interest and loan repayments (GP), modelling processes that grow linearly or exponentially, sums arising in combinatorics and probability, and analysis of algorithms.
• Understanding convergence is essential for infinite series used in calculus, power series and mathematical analysis.
• When working with sequences and series, always state the index range (where n starts) and check whether you are dealing with finite sums (series) or limits of partial sums (infinite series).

Summary - A sequence is an ordered list of terms defined by a rule; a series is the sum of terms of a sequence. Important types include arithmetic and geometric sequences, and special sequences such as the Fibonacci sequence. For series, formulas for sums of AP and GP and the idea of convergence are central. Practice computing n^th terms and partial sums, and apply tests for convergence when dealing with infinite series.