Complete Summary of Numerical Sequences – Definitions, Theorems and Formulas

Table of Contents

Numerical sequences are one of the most important topics in algebra and calculus. They are used to model population growth, financial investments, computer algorithms, engineering systems, physics, and many real-life phenomena. Every mathematics student should master the different types of sequences and the theorems associated with them.

Definition
A numerical sequence is a function defined on the set of natural numbers. Each natural number n corresponds to exactly one real number denoted by u_n.

1. Definition of a Numerical Sequence

A sequence is an ordered list of numbers:

u_{0}, u_{1}, u_{2}, u_{3}, \dots

Vocabulary

u₀ : First term
u₁ : Second term
uₙ : General term
n : Index (Rank)

2. Main Types of Numerical Sequences

Sequence Type	Description
Explicit Sequence	The nth term is given directly as a function of n.
Recursive Sequence	Each term depends on one or more previous terms.
Arithmetic Sequence	The difference between consecutive terms is constant.
Geometric Sequence	The ratio between consecutive terms is constant.

3. Explicit Sequence

Definition

u_{n} = f (n)

Each term can be calculated directly without knowing the previous terms.

Example

u_{n} = 4 n + 3

To calculate

u_{5}

simply substitute n = 5.

4. Recursive Sequence

Definition A recursive sequence is defined using one or more previous terms.

u_{n + 1} = f (u_{n})

Example

u_{0} = 2

u_{n + 1} = u_{n} + 5

Important Rule
To calculate a term, you must know the previous term(s).

5. Arithmetic Sequence

Definition An arithmetic sequence has a constant difference between consecutive terms.

Recurrence Formula

u_{n + 1} = u_{n} + r

r is called the common difference.

General Formula

u_{n} = u_{0} + n r

Theorem 1 An arithmetic sequence satisfies

u_{n + 1} - u_{n} = r

where r is a constant.

Property If

r > 0

the sequence is increasing.

If

r < 0

the sequence is decreasing.

6. Geometric Sequence

Definition
A sequence is called a geometric sequence when each term is obtained by multiplying the previous term by the same constant number called the common ratio.

Recursive Formula

u_{n + 1} = q u_{n}

Important Rule
q is called the common ratio.

Explicit Formula

u_{n} = u_{0} q^{n}

Theorem 2 A sequence is geometric if and only if

\frac{u_{n + 1}}{u_{n}} = q

where q is constant.

Property

If q > 1, the sequence is increasing (provided the terms are positive).
If 0 < q < 1, the sequence is decreasing.
If q = 1, the sequence is constant.
If q < 0, the signs alternate.

7. Sum of the First n Terms of an Arithmetic Sequence

Main Formula

S_{n} = \frac{n (u_{1} + u_{n})}{2}

Equivalent Formula

S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) r)

8. Sum of the First n Terms of a Geometric Sequence

Main Formula

S_{n} = \frac{u_{1} (1 - q^{n})}{1 - q}

Special Case If q = 1 then

S_{n} = n u_{1}

9. Box of Essential Formulas

Every Student Should Memorize

Arithmetic recurrence: $u_{n + 1} = u_{n} + r$
Arithmetic explicit formula: $u_{n} = u_{0} + n r$
Geometric recurrence: $u_{n + 1} = q u_{n}$
Geometric explicit formula: $u_{n} = u_{0} q^{n}$

10. Monotonicity of a Sequence

Definition
The monotonicity of a sequence describes how its terms evolve as the index increases.

Type of Sequence	Condition	Conclusion
Increasing	$u_{n + 1} > u_{n}$	The terms increase.
Decreasing	$u_{n + 1} < u_{n}$	The terms decrease.
Constant	$u_{n + 1} = u_{n}$	All terms are equal.

Theorem 3
If

u_{n + 1} - u_{n} > 0

then the sequence is increasing.

If

u_{n + 1} - u_{n} < 0

then the sequence is decreasing.

11. Bounded Sequences

Definition
A sequence is bounded if all its terms remain between two fixed real numbers.

m \leq u_{n} \leq M

Theorem 4
Every convergent sequence is bounded.

Important Remark
A bounded sequence is not necessarily convergent.

12. Convergent and Divergent Sequences

Definition
A sequence converges if its terms approach a fixed real number as the index tends to infinity.

lim_{n \to \infty} u_{n} = L

Theorem 5
A convergent sequence has a unique limit.

Examples

$u_{n} = \frac{1}{n}$ converges to 0.
$u_{n} = n$ diverges toward infinity.

13. Fundamental Theorems

Theorem 6 – Comparison Theorem

u_{n} \leq v_{n}

If both sequences converge, then their limits preserve the inequality.

Theorem 7 – Squeeze Theorem If

u_{n} \leq v_{n} \leq w_{n}

and

lim_{n \to \infty} u_{n} = lim_{n \to \infty} w_{n}

then

lim_{n \to \infty} v_{n}

has the same limit.

14. Summary Sheet

Essential Formulas Every Student Should Know

Arithmetic recurrence: $u_{n + 1} = u_{n} + r$
Arithmetic explicit: $u_{n} = u_{0} + n r$
Geometric recurrence: $u_{n + 1} = q u_{n}$
Geometric explicit: $u_{n} = u_{0} q^{n}$
Arithmetic sum: $S_{n} = \frac{n (u_{1} + u_{n})}{2}$
Geometric sum: $S_{n} = \frac{u_{1} (1 - q^{n})}{1 - q}$

Final Advice
To solve sequence problems successfully, always identify the type of sequence first (arithmetic, geometric, explicit, or recursive). Then apply the corresponding theorem and formula. Understanding the properties of monotonicity, boundedness, and convergence will make advanced topics in calculus and mathematical analysis much easier.

15. Sequence Limits

Definition
The limit of a sequence is the value approached by its terms as the index becomes infinitely large.

lim_{n \to \infty} u_{n} = L

Interpretation
As n becomes larger and larger, the terms of the sequence get arbitrarily close to the real number L.

16. Important Limits Every Student Should Know

Fundamental Limits

lim_{n \to \infty} \frac{1}{n} = 0

lim_{n \to \infty} n = + \infty

lim_{n \to \infty} \frac{1}{n^{2}} = 0

17. Sequence Classification

Sequence	Behavior	Limit
$\frac{1}{n}$	Decreasing	0
n	Increasing	+∞
$2^{n}$	Very fast increasing	+∞
$\frac{1}{2^{n}}$	Very fast decreasing	0

18. Mathematical Proof by Induction

Principle of Mathematical Induction To prove that a property is true for every natural number, follow these three steps.

Step	Description
1	Verify the property for the first value.
2	Assume the property is true for n.
3	Prove it remains true for n+1.

Notation

P (n)

represents the mathematical property to be proved.

19. Common Mistakes

Confusing the common difference with the common ratio.
Using the arithmetic formula for a geometric sequence.
Forgetting the initial term.
Ignoring the index where the sequence begins.
Using the wrong summation formula.
Assuming every bounded sequence converges.
Applying induction incorrectly.

20. Complete Formula Sheet

Arithmetic Sequence

u_{n} = u_{0} + n r

Geometric Sequence

u_{n} = u_{0} q^{n}

Arithmetic Sum

S_{n} = \frac{n (u_{1} + u_{n})}{2}

Geometric Sum

S_{n} = \frac{u_{1} (1 - q^{n})}{1 - q}

Final Summary
To solve any sequence problem efficiently:

Identify the type of sequence.
Determine whether it is explicit or recursive.
Apply the correct theorem.
Choose the appropriate formula.
Simplify carefully.
Verify monotonicity and convergence when required.

Students who master these concepts will be well prepared for algebra, calculus, mathematical analysis, engineering, economics, and university entrance examinations.

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21. Solved Exercises

Exercise 1 – Arithmetic Sequence

Given the arithmetic sequence:

u_{0} = 5, r = 3

Find u₁₀.

Solution

u_{n} = u_{0} + n r

u_{10} = 5 + 10 \times 3 = 35

Exercise 2 – Geometric Sequence

Given

u_{0} = 2, q = 4

Calculate u₅.

u_{n} = u_{0} q^{n}

u_{5} = 2 4^{5} = 2048

22. Theorems Every Student Must Know

Theorem 8 A sequence that is both increasing and bounded above is convergent.

Theorem 9 A sequence that is decreasing and bounded below is convergent.

Theorem 10 Every geometric sequence satisfies

u_{n} = u_{p} q^{n - p}

for every pair of integers p and n.

Theorem 11 Every arithmetic sequence satisfies

u_{n} = u_{p} + (n - p) r

23. Comparison Between Arithmetic and Geometric Sequences

Property	Arithmetic	Geometric
Operation	Addition	Multiplication
Constant	Difference (r)	Ratio (q)
Explicit Formula	u₀ + nr	u₀qⁿ
Growth	Linear	Exponential
Applications	Salary increase, temperature	Population, finance, bacteria

Memory Rule

Arithmetic → Add the same number.
Geometric → Multiply by the same number.

24. Formula Box