Overview: Progressions (Sequences & Series)

Table of Contents
1. Introduction
2. Sequence and Series - Notation and Definition
3. Common Types of Sequences and Series
4. Sequence and Series - Key Formulas
5. Difference Between Sequence and Series
6. Sequence and Series Examples
7. Useful Remarks and Problem-solving Tips
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Introduction

Sequence and series are fundamental topics in arithmetic and quantitative aptitude. A sequence is an ordered list of numbers (or objects) that follow a specific rule. A series is the sum of the terms of a sequence. Common examples include arithmetic progressions and geometric progressions, which are widely used in problem solving for competitive examinations.

• A sequence is a list of items arranged in a definite order.
• A series is the sum of terms of a sequence; the sequence must have a definite relationship between terms.

Sequence and Series - Notation and Definition

Let the terms of a sequence be denoted by a₁, a₂, a₃, a₄, ... where the index 1, 2, 3, 4, ... denotes the position of each term.

A sequence may be finite or infinite. If a₁, a₂, ..., a_N is a finite sequence, the corresponding finite series is

S_N = a₁ + a₂ + a₃ + ... + a_N

If the sequence is infinite, the associated series may be finite (convergent) or infinite (divergent) depending on whether the infinite sum converges.

In CAT-oriented problems, infinite series-when discussed-are typically limited to geometric progressions with |r| < 1.

Common Types of Sequences and Series

• Arithmetic sequences (Arithmetic Progression, AP)
• Geometric sequences (Geometric Progression, GP)
• Harmonic sequences (HP)
• Fibonacci sequence

Arithmetic Progression (AP)

An arithmetic progression is a sequence in which each term after the first is obtained by adding a fixed number, called the common difference d, to the previous term. If a is the first term, the sequence is

a, a + d, a + 2d, a + 3d, ...

The nth term (T_n) of an AP is

T_n = a + (n - 1)d

The last term l (if there are n terms) satisfies

l = a + (n - 1)d

The sum of the first n terms (S_n) of an AP is

S_n = n/2 × [2a + (n - 1)d] = n/2 × (a + l)

Geometric Progression (GP)

A geometric progression is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed number, called the common ratio r. If a is the first term, the sequence is

a, ar, ar², ar³, ...

The nth term (T_n) of a GP is

T_n = a r^n-1

The sum of the first n terms (S_n) for r ≠ 1 is

S_n = a × (1 - rⁿ) / (1 - r)

If |r| < 1, the infinite geometric series converges and its sum is

S_∞ = a / (1 - r)

Note: In CAT, infinite series questions are limited only to geometric progressions with |r| < 1. 

Harmonic Progression (HP)

A sequence is in harmonic progression if the reciprocals of its terms form an arithmetic progression. For example, if b_n = 1/a_n is an AP, then a_n is an HP.

If a₁, a₂, a_{3,...... are in HP, then 1/a1}, _1/a2, _1/a3, ...... are in AP. 

Note: Problems involving HP are solved by converting the HP into an AP by taking reciprocals of the terms.  

Fibonacci Sequence

The Fibonacci sequence is defined by the recurrence

F₀ = 0, F₁ = 1, and F_n = F_n-1 + F_n-2 for n ≥ 2.

The sequence begins 0, 1, 1, 2, 3, 5, 8, 13, ...

Note: For CAT, Fibonacci problems are solved using the recurrence relation or pattern recognition. Closed-form formulas are not required. 

Sequence and Series - Key Formulas

*Here, a = first term, d = common difference, r = common ratio, n = position of term, l = last term.

• AP nth term: T_n = a + (n - 1)d
• AP sum: S_n = n/2 × [2a + (n - 1)d] = n/2 × (a + l)
• GP nth term: T_n = a r^n-1
• GP sum (r ≠ 1): S_n = a × (1 - rⁿ) / (1 - r)
• GP infinite sum (|r| < 1): S_∞ = a / (1 - r)
• HP: a_n such that 1/a_n is arithmetic
• Fibonacci recurrence: F_n = F_n-1 + F_n-2

Difference Between Sequence and Series

A sequence lists elements in order; a series sums elements of a sequence. A sequence may be finite or infinite. A series formed from an infinite sequence may converge (sum has a finite value) or diverge.

Sequence and Series Examples

Question 1: If 4,7,10,13,16,19,22......is a sequence, Find:
(a) Common difference
(b) nth term
(c) 21st term
Sol: 

(a) The common difference = 7 - 4 = 3
(b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by T_n = a + (n-1)d, where "a" is the first term and d is the common difference.
T_n = 4 + (n - 1)3 = 4 + 3n - 3 = 3n + 1
(c) 21st term as:  T₂₁ = 4 + (21-1)3 = 4+60 = 64.

Q2: Consider the sequence 1, 4, 16, 64, 256, 1024..... Find the common ratio and 9th term.
Sol: The common ratio (r)  = 4/1 = 4
The preceding term is multiplied by 4 to obtain the next term.
The nth term of the geometric sequence is denoted by the term Tn and is given by Tn = ar(n-1) where a is the first term and r is the common ratio.
Here a = 1, r = 4 and n = 9
So, 9th term is can be calculated as T9 = 1* (4)^(9-1)= 4⁸ = 65536.

Useful Remarks and Problem-solving Tips

• To identify an AP, check for constant differences between consecutive terms. To identify a GP, check for constant ratios.
• When asked for the nth term, substitute a and d (or r) into the relevant nth-term formula.
• When given sum-related conditions, express the sum using S_n formulas and solve for unknowns.
• For infinite GP sums, ensure |r| < 1 before using S_∞ = a/(1 - r).
• Recurrence relations (like Fibonacci) often require initial terms; use them to generate further terms or derive closed forms when needed.