Σ
Terminale Mathematics
Numerical
Sequences
Suites Numériques
A Complete Guide from A to Z
Baccalauréat Level · Terminal Year
What you will master:
Arithmetic Sequences · Geometric Sequences
Recursive Sequences · Monotonicity & Bounds
Limits · Convergence · Series Sums
Applications to Finance & Real-World Problems
Author: ZAYD gg
store of BIGBOSSES
Academic Year 20242025
, About This Book
Terminale
Baccalauréat
This course has been carefully crafted for (Year 13)
students preparing for the . Every concept is
built from scratch with full mathematical rigour, clear intuition,
worked examples, mini exercises, and powerful exam strategies.
A sequence is a function in disguise
and every function tells a story.
Author: Publisher:
Level:
ZAYD gg BIGBOSSES
Subject:
Terminale Baccalauréat
Mathematics Suites Numériques
, Contents
1 Introduction to Sequences 5
1.1 What Is a Sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Two Ways to De
ne a Sequence . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Arithmetic Sequences 7
2.1 De
nition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Sum of an Arithmetic Sequence . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Geometric Sequences 11
3.1 De
nition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Sum of a Geometric Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Monotonicity and Boundedness 15
4.1 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Bounded Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Limits of Sequences 17
5.1 Intuition and De
nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Standard Limits to Know . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3 Limit Rules and Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4 Factorisation Technique for Limits . . . . . . . . . . . . . . . . . . . . . . . 18
6 Convergence Theorems 21
6.1 The Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . 22
7 Recursive Sequences and Fixed Points 23
7.1 Sequences of the Form un+1 = f (un ) . . . . . . . . . . . . . . . . . . . . . 23
7.2 A
ne Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8 Geometric Series and Financial Applications 25
8.1 Sum of an In
nite Geometric Series . . . . . . . . . . . . . . . . . . . . . . 25
8.2 Financial Mathematics Capital and Interest . . . . . . . . . . . . . . . . 25
9 Comparing Growth Rates 27
9.1 The Hierarchy of Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
9.2 Comparison of Sequences and Equivalent Sequences . . . . . . . . . . . . . 27
3
,Numerical Sequences Suites Numériques CONTENTS
10 Key Formulas and Master Sheet 29
10.1 Complete Formula Reference . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10.2 Standard Argument Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
11 Full Practice Exercises 31
12 Detailed Solutions 33
Solutions to Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Solutions to Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13 Exam Tips and Strategy 35
13.1 Top 10 Things to Know for the Bac . . . . . . . . . . . . . . . . . . . . . . 35
13.2 7 Classic Mistakes to Avoid . . . . . . . . . . . . . . . . . . . . . . . . . . 35
13.3 Rapid Revision: 3-Minute Checklist . . . . . . . . . . . . . . . . . . . . . . 36
14 Supplementary Bac-Style Problems 37
15 Induction and Sequences 39
15.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
15.2 Strong Induction and the Fibonacci Sequence . . . . . . . . . . . . . . . . 40
16 Advanced Limit Techniques 41
16.1 Limits via Substitution and Change of Index . . . . . . . . . . . . . . . . . 41
16.2 Cesàro Mean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
17 Special Sequences and Classic Results
n
43
17.1 The Sequence (1 + 1/n) → e . . . . . . . . . . . . . . . . . . . . . . . . . 43
17.2 Harmonic Series and Divergence . . . . . . . . . . . . . . . . . . . . . . . . 43
17.3 Sequences De
ned by an Integral . . . . . . . . . . . . . . . . . . . . . . . 44
18 Complete Summary and Visual Guide 45
18.1 Decision Flowchart: What Type Is My Sequence? . . . . . . . . . . . . . . 46
18.2 Visual Summary of Convergence and Divergence . . . . . . . . . . . . . . . 46
18.3 The Four Key Theorems at a Glance . . . . . . . . . . . . . . . . . . . . . 47
19 Glossary and Index of Key Terms 49
4 ZAYD gg | BIGBOSSES
, CHAPTER 1
Introduction to Sequences
1.1 What Is a Sequence?
In everyday life we encounter ordered lists: the temperature each day, the balance of a
sequence
bank account month by month, the position of a moving object at each second. Mathe-
matics formalises this idea with the concept of a .
❖ De
nition Numerical Sequence
numerical sequence (u )
general term n
A n n∈N is a function from N (or a subset of N) to R. The
value u n is called the (or -th term) of the sequence.
The index n usually starts at 0 or 1 (always speci
ed).
We write (un ), (un )n≥0 , or (un )n≥1 .
un
(un )
n
1 2 3 4 5 6
1.2 Two Ways to De
ne a Sequence
Explicit Formula vs. Recursive Formula
1. Explicit (closed-form) formula: u n is given directly as a function of n.
Example: un = 3n + 1 ⇒ u0 = 1, u1 = 4, u2 = 7, . . .
2. Recursive formula (recurrence relation): u n+1 is expressed in terms of un
5
,Numerical Sequences Suites Numériques
CHAPTER 1. INTRODUCTION TO SEQUENCES
(and possibly earlier terms), together with an initial condition .
Example: u0 = 1, un+1 = 2un + 1 ⇒ u0 = 1, u1 = 3, u2 = 7, u3 = 15, . . .
✩ Key Trick Which Formula to Prefer?
Use explicit formulas to compute any term directly (no need to compute all
previous terms).
Use recursive formulas when the sequence is de
ned by a physical or
nancial
process.
nd the
explicit formula
On the Bac exam: if you have a recursive formula, always try to
it makes computing limits and sums much easier!
✎ Mini Exercise 1.1 First Terms
For each sequence, write the
rst
ve terms (n = 0, 1, 2, 3, 4):
1. u = n − 2n + 1
n
2
2. v = (−1)
n
n+1
n
3. w = 3 w = w2 + 1
0 , n+1
n
4. t = 1 t = 1 t = t
1 , 2 , n+2 n+1 + tn (Fibonacci!)
Answers: 1, 0, 1, 4, 9
(1) (2) 1, − 21 , 13 , − 41 , 15 (3) 3, 2.5, 2.25, 2.125, 2.0625 (4)
1, 1, 2, 3, 5
6 ZAYD gg | BIGBOSSES
, CHAPTER 2
Arithmetic Sequences
2.1 De
nition and Properties
❖ De
nition Arithmetic Sequence
A sequence (u ) arithmetic
n is common dierence r ∈ R
with if:
un+1 = un + r for all n≥0
Equivalently: un+1 − un = r (constant dierence between consecutive terms).
If r > 0: sequence is increasing
If r < 0: sequence is decreasing
If r = 0: sequence is constant
un r
r = 0.7
r
r
r
r
r
n
General Term of an Arithmetic Sequence
If (un ) is arithmetic with
rst term u0 (or u1 ) and common dierence r:
un = u0 + nr or un = u1 + (n − 1)r
More generally, for any two indices p and q:
un = up + (n − p)r
7
, Numerical Sequences Suites NumériquesCHAPTER 2. ARITHMETIC SEQUENCES
✩ Key Trick Identifying an Arithmetic Sequence
To check if (un ) is arithmetic:
1. Compute un+1 − un .
2. If the result is a constant (independent of n) ⇒ arithmetic with r = un+1 − un .
Example: u n = 5n − 3. Then un+1 − un = 5(n + 1) − 3 − (5n − 3) = 5. Arithmetic
with r = 5.
2.2 Sum of an Arithmetic Sequence
★ Sum of Arithmetic Terms
The sum of the
rst n+1 terms (from u0 to un ) of an arithmetic sequence:
n
X u0 + un
Sn = uk = (n + 1) ·
k=0
2
In words: number of terms × average of
rst and last term.
Special case (sum 1 + 2 + · · · + n):
n
X n(n + 1)
k=
k=1
2
✩ Memory Trick The Gauss Story
Young Gauss (age 9) was asked to sum 1 + 2 + · · · + 100. He paired them: (1 + 100) +
(2 + 99) + · · · + (50 + 51) = 50 × 101 = 5050. This is exactly the formula: n(n + 1)
2
with n = 100 gives 5050.
Worked Example:
X 10
The terms of an arithmetic sequence satisfy u3 = 11 and u7 = 27.
Find r, u0 , and uk .
Solution: u − u k=0
= 4r ⇒ 27 − 11 = 4r ⇒ r = 4. u0 = u3 − 3r = 11 − 12 = −1.
7 3
10
X u0 + u10 −1 + 39
u10 = −1 + 10 × 4 = 39. S10 = uk = 11 × = 11 × = 11 × 19 = 209.
k=0
2 2
✎ Mini Exercise 2.1 Arithmetic Sequences
1. Show that un = 7 − 3n is arithmetic. Give r and u0 .
2. An arithmetic sequence has u1 = 5 and u5 = 21. Find r, u10 , and
P10
k=1 uk .
3. Compute: 3 + 7 + 11 + · · · + 99 (arithmetic, r = 4).
4. The sum of the
rst n terms of an arithmetic sequence is Sn = 2n2 + 3n. Find
8 ZAYD gg | BIGBOSSES
Terminale Mathematics
Numerical
Sequences
Suites Numériques
A Complete Guide from A to Z
Baccalauréat Level · Terminal Year
What you will master:
Arithmetic Sequences · Geometric Sequences
Recursive Sequences · Monotonicity & Bounds
Limits · Convergence · Series Sums
Applications to Finance & Real-World Problems
Author: ZAYD gg
store of BIGBOSSES
Academic Year 20242025
, About This Book
Terminale
Baccalauréat
This course has been carefully crafted for (Year 13)
students preparing for the . Every concept is
built from scratch with full mathematical rigour, clear intuition,
worked examples, mini exercises, and powerful exam strategies.
A sequence is a function in disguise
and every function tells a story.
Author: Publisher:
Level:
ZAYD gg BIGBOSSES
Subject:
Terminale Baccalauréat
Mathematics Suites Numériques
, Contents
1 Introduction to Sequences 5
1.1 What Is a Sequence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Two Ways to De
ne a Sequence . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Arithmetic Sequences 7
2.1 De
nition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Sum of an Arithmetic Sequence . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Geometric Sequences 11
3.1 De
nition and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Sum of a Geometric Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Monotonicity and Boundedness 15
4.1 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Bounded Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Limits of Sequences 17
5.1 Intuition and De
nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Standard Limits to Know . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.3 Limit Rules and Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.4 Factorisation Technique for Limits . . . . . . . . . . . . . . . . . . . . . . . 18
6 Convergence Theorems 21
6.1 The Squeeze Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2 Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . . . . . 22
7 Recursive Sequences and Fixed Points 23
7.1 Sequences of the Form un+1 = f (un ) . . . . . . . . . . . . . . . . . . . . . 23
7.2 A
ne Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
8 Geometric Series and Financial Applications 25
8.1 Sum of an In
nite Geometric Series . . . . . . . . . . . . . . . . . . . . . . 25
8.2 Financial Mathematics Capital and Interest . . . . . . . . . . . . . . . . 25
9 Comparing Growth Rates 27
9.1 The Hierarchy of Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
9.2 Comparison of Sequences and Equivalent Sequences . . . . . . . . . . . . . 27
3
,Numerical Sequences Suites Numériques CONTENTS
10 Key Formulas and Master Sheet 29
10.1 Complete Formula Reference . . . . . . . . . . . . . . . . . . . . . . . . . . 29
10.2 Standard Argument Table . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
11 Full Practice Exercises 31
12 Detailed Solutions 33
Solutions to Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Solutions to Exercise 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Solutions to Exercise 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
13 Exam Tips and Strategy 35
13.1 Top 10 Things to Know for the Bac . . . . . . . . . . . . . . . . . . . . . . 35
13.2 7 Classic Mistakes to Avoid . . . . . . . . . . . . . . . . . . . . . . . . . . 35
13.3 Rapid Revision: 3-Minute Checklist . . . . . . . . . . . . . . . . . . . . . . 36
14 Supplementary Bac-Style Problems 37
15 Induction and Sequences 39
15.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
15.2 Strong Induction and the Fibonacci Sequence . . . . . . . . . . . . . . . . 40
16 Advanced Limit Techniques 41
16.1 Limits via Substitution and Change of Index . . . . . . . . . . . . . . . . . 41
16.2 Cesàro Mean Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
17 Special Sequences and Classic Results
n
43
17.1 The Sequence (1 + 1/n) → e . . . . . . . . . . . . . . . . . . . . . . . . . 43
17.2 Harmonic Series and Divergence . . . . . . . . . . . . . . . . . . . . . . . . 43
17.3 Sequences De
ned by an Integral . . . . . . . . . . . . . . . . . . . . . . . 44
18 Complete Summary and Visual Guide 45
18.1 Decision Flowchart: What Type Is My Sequence? . . . . . . . . . . . . . . 46
18.2 Visual Summary of Convergence and Divergence . . . . . . . . . . . . . . . 46
18.3 The Four Key Theorems at a Glance . . . . . . . . . . . . . . . . . . . . . 47
19 Glossary and Index of Key Terms 49
4 ZAYD gg | BIGBOSSES
, CHAPTER 1
Introduction to Sequences
1.1 What Is a Sequence?
In everyday life we encounter ordered lists: the temperature each day, the balance of a
sequence
bank account month by month, the position of a moving object at each second. Mathe-
matics formalises this idea with the concept of a .
❖ De
nition Numerical Sequence
numerical sequence (u )
general term n
A n n∈N is a function from N (or a subset of N) to R. The
value u n is called the (or -th term) of the sequence.
The index n usually starts at 0 or 1 (always speci
ed).
We write (un ), (un )n≥0 , or (un )n≥1 .
un
(un )
n
1 2 3 4 5 6
1.2 Two Ways to De
ne a Sequence
Explicit Formula vs. Recursive Formula
1. Explicit (closed-form) formula: u n is given directly as a function of n.
Example: un = 3n + 1 ⇒ u0 = 1, u1 = 4, u2 = 7, . . .
2. Recursive formula (recurrence relation): u n+1 is expressed in terms of un
5
,Numerical Sequences Suites Numériques
CHAPTER 1. INTRODUCTION TO SEQUENCES
(and possibly earlier terms), together with an initial condition .
Example: u0 = 1, un+1 = 2un + 1 ⇒ u0 = 1, u1 = 3, u2 = 7, u3 = 15, . . .
✩ Key Trick Which Formula to Prefer?
Use explicit formulas to compute any term directly (no need to compute all
previous terms).
Use recursive formulas when the sequence is de
ned by a physical or
nancial
process.
nd the
explicit formula
On the Bac exam: if you have a recursive formula, always try to
it makes computing limits and sums much easier!
✎ Mini Exercise 1.1 First Terms
For each sequence, write the
rst
ve terms (n = 0, 1, 2, 3, 4):
1. u = n − 2n + 1
n
2
2. v = (−1)
n
n+1
n
3. w = 3 w = w2 + 1
0 , n+1
n
4. t = 1 t = 1 t = t
1 , 2 , n+2 n+1 + tn (Fibonacci!)
Answers: 1, 0, 1, 4, 9
(1) (2) 1, − 21 , 13 , − 41 , 15 (3) 3, 2.5, 2.25, 2.125, 2.0625 (4)
1, 1, 2, 3, 5
6 ZAYD gg | BIGBOSSES
, CHAPTER 2
Arithmetic Sequences
2.1 De
nition and Properties
❖ De
nition Arithmetic Sequence
A sequence (u ) arithmetic
n is common dierence r ∈ R
with if:
un+1 = un + r for all n≥0
Equivalently: un+1 − un = r (constant dierence between consecutive terms).
If r > 0: sequence is increasing
If r < 0: sequence is decreasing
If r = 0: sequence is constant
un r
r = 0.7
r
r
r
r
r
n
General Term of an Arithmetic Sequence
If (un ) is arithmetic with
rst term u0 (or u1 ) and common dierence r:
un = u0 + nr or un = u1 + (n − 1)r
More generally, for any two indices p and q:
un = up + (n − p)r
7
, Numerical Sequences Suites NumériquesCHAPTER 2. ARITHMETIC SEQUENCES
✩ Key Trick Identifying an Arithmetic Sequence
To check if (un ) is arithmetic:
1. Compute un+1 − un .
2. If the result is a constant (independent of n) ⇒ arithmetic with r = un+1 − un .
Example: u n = 5n − 3. Then un+1 − un = 5(n + 1) − 3 − (5n − 3) = 5. Arithmetic
with r = 5.
2.2 Sum of an Arithmetic Sequence
★ Sum of Arithmetic Terms
The sum of the
rst n+1 terms (from u0 to un ) of an arithmetic sequence:
n
X u0 + un
Sn = uk = (n + 1) ·
k=0
2
In words: number of terms × average of
rst and last term.
Special case (sum 1 + 2 + · · · + n):
n
X n(n + 1)
k=
k=1
2
✩ Memory Trick The Gauss Story
Young Gauss (age 9) was asked to sum 1 + 2 + · · · + 100. He paired them: (1 + 100) +
(2 + 99) + · · · + (50 + 51) = 50 × 101 = 5050. This is exactly the formula: n(n + 1)
2
with n = 100 gives 5050.
Worked Example:
X 10
The terms of an arithmetic sequence satisfy u3 = 11 and u7 = 27.
Find r, u0 , and uk .
Solution: u − u k=0
= 4r ⇒ 27 − 11 = 4r ⇒ r = 4. u0 = u3 − 3r = 11 − 12 = −1.
7 3
10
X u0 + u10 −1 + 39
u10 = −1 + 10 × 4 = 39. S10 = uk = 11 × = 11 × = 11 × 19 = 209.
k=0
2 2
✎ Mini Exercise 2.1 Arithmetic Sequences
1. Show that un = 7 − 3n is arithmetic. Give r and u0 .
2. An arithmetic sequence has u1 = 5 and u5 = 21. Find r, u10 , and
P10
k=1 uk .
3. Compute: 3 + 7 + 11 + · · · + 99 (arithmetic, r = 4).
4. The sum of the
rst n terms of an arithmetic sequence is Sn = 2n2 + 3n. Find
8 ZAYD gg | BIGBOSSES
