8. Sequences and Series – Class 11 Mathematics

Introduction

The study of sequences and series lies at the heart of mathematics because it combines logical reasoning with algebraic manipulation. Many natural phenomena such as population growth, compound interest, radioactive decay, and even computer algorithms rely on patterns that can be modeled using sequences and series.

In NCERT Class 11 Chapter 8, we learn about:

• How sequences are formed
• How to represent and analyze them
• The difference between a sequence and a series
• Arithmetic and geometric progressions (A.P. and G.P.)
• The inequality relationship between A.M. (Arithmetic Mean) and G.M. (Geometric Mean)

This topic is particularly useful in competitive exams like JEE, KCET, and COMEDK, where questions often test formula application, problem-solving speed, and conceptual clarity.

8.2 Sequences

Definition of a Sequence

A sequence is an ordered list of numbers formed according to a definite rule. If the nth term of the sequence is represented by an, then the sequence is written as:

{a1, a2, a3, …, an, …}

Examples:

1. Natural numbers: 1, 2, 3, 4, …
2. Squares: 1, 4, 9, 16, … (where an = n²)
3. Powers of 2: 2, 4, 8, 16, … (where an = 2n)

Types of Sequences

1. Finite Sequence – Ends after a certain number of terms.
   Example: 2, 4, 6, 8 (only 4 terms). 
2. Infinite Sequence – Goes on endlessly.
   Example: 1, 1/2, 1/3, 1/4, … 
3. Arithmetic Sequence (A.P.) – Each term differs by a constant difference d.
   Example: 3, 6, 9, 12, … (d = 3). 
4. Geometric Sequence (G.P.) – Each term is obtained by multiplying the previous one by a constant ratio r.
   Example: 2, 6, 18, 54, … (r = 3). 
5. Harmonic Sequence (H.P.) – The reciprocals of terms form an A.P.
   Example: 1, 1/2, 1/3, 1/4, … 

8.3 Series

Definition of a Series

When the terms of a sequence are added, we get a series.

For a sequence {an}, the series is:
S = a1 + a2 + a3 + … + an

Types of Series

• Arithmetic Series: Sum of an arithmetic progression.
• Geometric Series: Sum of a geometric progression.
• Harmonic Series: Sum of a harmonic progression.

Important Formulas

1. Sum of first n terms of an A.P.
   Sn = n/2 [2a + (n – 1)d]
2. Sum of first n terms of a G.P.
   Sn = a(1 – rn)/(1 – r), r ≠ 1
3. Sum to infinity of a G.P.
   S∞ = a/(1 – r), if |r| < 1

8.4 Geometric Progression (G.P.)

Definition

A G.P. is a sequence of the form:
a, ar, ar², ar³, …

where a is the first term and r is the common ratio.

Important Formulas

• nth term: Tn = ar(n–1)
• Sum of n terms: Sn = a(1 – rn)/(1 – r), r ≠ 1
• Sum to infinity: S∞ = a/(1 – r), |r| < 1

Real-life Applications

• Compound interest calculations
• Population growth
• Radioactive decay
• Computer science (binary growth, algorithms)

8.5 Relationship Between A.M. and G.M.

For two positive numbers a and b:

• Arithmetic Mean (A.M.) = (a + b)/2
• Geometric Mean (G.M.) = √(ab)

Key Inequality

A.M. ≥ G.M.
Equality holds only when a = b.

This inequality is highly important for solving optimization problems and proving inequalities in JEE Advanced.

Solved Examples

Example 1 (A.P. nth term)

Find the 15th term of the sequence 5, 8, 11, …

Solution:
a = 5, d = 3
Tn = a + (n – 1)d
T15 = 5 + (15 – 1) × 3 = 5 + 42 = 47

Example 2 (A.P. Sum)

Find the sum of the first 20 terms of the sequence 2, 4, 6, …

Solution:
a = 2, d = 2, n = 20
Sn = n/2 [2a + (n – 1)d]
= 20/2 [4 + (19 × 2)]
= 10 × (4 + 38) = 420

Example 3 (G.P. nth term)

Find the 8th term of the sequence 3, 6, 12, …

Solution:
a = 3, r = 2
Tn = ar(n–1)
T8 = 3 × 2(7) = 384

Example 4 (Sum of G.P.)

Find the sum of the first 6 terms of the G.P. 2, 4, 8, 16, …

Solution:
a = 2, r = 2, n = 6
Sn = a(1 – rn)/(1 – r)
= 2(1 – 26)/(1 – 2)
= 2(1 – 64)/(–1) = 126

Example 5 (A.M.–G.M. Inequality)

If A = 4 and B = 9, find A.M. and G.M.

Solution:
A.M. = (4 + 9)/2 = 6.5
G.M. = √(36) = 6
So A.M. > G.M.

Example 6 (Application – Compound Interest)

If Rs. 5000 is invested at 10% annual compound interest, find the amount after 3 years.

Solution:
Formula: A = P(1 + r)n
= 5000(1 + 0.1)3
= 5000(1.331) = Rs. 6655

This uses the formula for G.P. sum.

Applications in Exams

JEE (Main and Advanced)

• Problems on the nth term and sum of A.P. and G.P.
• Proofs involving A.M.–G.M. inequality.
• Infinite series and convergence.

KCET/COMEDK

• Formula-based MCQs.
• Direct numerical questions on nth term and sum of n terms.

CBSE/PU Boards

• 8–10 mark long-answer questions involving step-by-step derivations.

Marks Allotment

• CBSE Boards: 8–10 marks
• JEE Main: 4–8 marks (direct formula, sums of series)
• KCET/COMEDK: 6–8 marks (MCQs and quick sums)
• JEE Advanced: 6–10 marks (indirect application in inequalities, convergence, multiple series problems)

Common Pitfalls

1. Forgetting to use the correct formula for Sn when r = 1 in G.P.
2. Mixing up A.M. ≥ G.M. with equality cases.
3. Careless mistakes in exponentiation in G.P. problems.
4. Wrong application of formulas for finite vs infinite series.

Practice Problems

1. Find the 25th term of the sequence 7, 10, 13, …
2. Find the sum of the first 50 terms of 5 + 10 + 15 + …
3. Find the sum of the first 8 terms of the G.P. 1, 2, 4, 8, …
4. If a = 2, b = 8, find A.M. and G.M.
5. A man saves Rs. 100 in the first month, Rs. 200 in the second month, Rs. 300 in the third month, and so on. Find the total savings after 12 months.

FAQs

Q1. What is the difference between sequence and series?
A sequence lists numbers in order, while a series is the sum of those numbers.

Q2. What is the sum to infinity of a G.P.?
If |r| < 1, S∞ = a/(1 – r).

Q3. Why is A.M. ≥ G.M.?
Because (a – b)² ≥ 0 always holds, proving (a + b)/2 ≥ √(ab).

Q4. How many marks are allotted for this chapter in JEE?
Around 4–8 marks in JEE Main and up to 10 marks in JEE Advanced.

Q5. What is the real-life use of G.P.?
It is widely used in finance (compound interest), biology (population growth), and physics (radioactive decay).