The document provides an overview of sequences and series, defining them as ordered lists of numbers and their sums, respectively, along with examples and types such as arithmetic and geometric sequences. It includes formulas for finding terms and sums, as well as applications in finance, physics, and computer science. The importance of these mathematical concepts and their real-world significance are emphasized, encouraging further exploration.

1. Sequences and Series
Students Name:
Faisal Buradha 224009217
Nawaf Naif Alghanim 224009114
Abdulrahman Almulhim: 224015805
Abdulaziz Alfuwaires 224011618
Abdulhadi almarhabi 224013200 Supervised by
Dr. Muhammad Nur

2. Introduction
What are Sequences?
Definition: An ordered list of numbers following a specific pattern or rule.
Examples: 2, 4, 6, 8... OR 1, 1, 2, 3, 5, 8...
What are Series?
Definition: The sum of the terms in a sequence.
Examples: 2 + 4 + 6 + 8 + ... OR 1 + 1 + 2 + 3 + 5 + 8 + ...
Importance: Sequences and series are fundamental in mathematics and have
applications in various fields like finance, physics, and computer science.

3. Types of Sequences
• Arithmetic Sequences:
• Definition: Each term is obtained by adding a constant value (common difference) to the previous term.
• Formula: an = a1 + (n-1)d
• an: nth term
• a1: first term
• d: common difference
• Example: 3, 7, 11, 15... (d = 4)
• Geometric Sequences:
• Definition: Each term is obtained by multiplying the previous term by a constant value (common ratio).
• Formula: an = a1 * r^(n-1)
• an: nth term
• a1: first term
• r: common ratio
• Example: 2, 6, 18, 54... (r = 3)
• Other Sequences: Fibonacci sequence, harmonic sequence

4. Finding the nth Term
•Arithmetic Sequences: Reiterate the formula and
provide an example problem: "Find the 10th term of
the sequence: 5, 8, 11,
14..."Geometric Sequences: Reiterate the formula
and provide an example problem: "Find the 6th term
of the sequence: 1, 3, 9, 27..."

5. Arithmetic Series
• Definition: The sum of an arithmetic sequence.
• Formulae:
• Sn = (n/2) * (a1 + an)
• Sn = (n/2) * [2a1 + (n-1)d]
• Sn: sum of the first n terms
• Example Problem: Find the sum of the first 20 terms of the
sequence: 2, 5, 8, 11...

6. Geometric Series
• Definition: The sum of a geometric sequence.
• Formula:
• Sn = a1 * (1 - rn) / (1 - r)
• Sn: sum of the first n terms
• Example Problem: Find the sum of the first 8 terms of the sequence:
1, 2, 4, 8...

7. Infinite Geometric Series
• Definition: A geometric series with an infinite number of terms.
• Convergence: An infinite geometric series converges (has a finite
sum) only if the absolute value of the common ratio (|r|) is less than
1.
• Formula:
• S = a1 / (1 - r) (if |r| < 1)
• S: sum of the infinite series
• Example Problem: Find the sum of the infinite series: 1 + 1/2 + 1/4 +
1/8...

8. Sigma Notation
• Introduction: A shorthand way to represent sums, especially for
series.
• Explanation: Σ (sigma) symbol represents summation.
• Example: Σn=15 n2 = 12 + 22 + 32 + 42 + 52

9. Applications of Sequences and Series
• Finance: Compound interest, annuities.
• Physics: Motion of falling objects, oscillations.
• Computer Science: Algorithms, data analysis.
• Nature: Fibonacci sequence in plant growth, population growth
models.

10. Visualizations
• Include visually engaging diagrams or graphs:
• Graph of an arithmetic and geometric sequence to illustrate their growth.
• Diagram showing the convergence of an infinite geometric series.
• Visual representation of the Fibonacci sequence in nature (e.g., spiral in a
seashell).

11. Conclusion
Recap the key concepts of sequences and series.
Emphasize their importance and real-
world applications.
Encourage further exploration and learning about this
fascinating area of mathematics

12. references
1.Khan Academy (https://www.khanacademy.org/)
2.MathWorld (https://mathworld.wolfram.com/)

13. Thank you