03-07 - Sequences and Series
Phase: 3 | Subject: 03-07 Prerequisites: 01-09-polynomials.md (polynomial manipulation) Next subject: 03-08-mathematical-induction.md
Learning Objectives
By the end of this subject, you will be able to:
- Define arithmetic and geometric sequences
- Find the n-th term formula for each type
- Calculate sums of finite arithmetic and geometric series
- Determine convergence of infinite geometric series
- Use sigma notation for sums
- Define and compute terms of recursive sequences
Core Content
Sequences
A sequence is an ordered list of numbers. Each number is a term.
Example: 2, 5, 8, 11, 14, ... (arithmetic, common difference 3)
Arithmetic Sequences
Each term differs by a constant common difference d.
n-th term: aₙ = a₁ + (n - 1)d
Example: 3, 7, 11, 15, ... a₁ = 3, d = 4 aₙ = 3 + (n - 1)(4) = 4n - 1
Sum of first n terms: Sₙ = (n/2)(a₁ + aₙ) = (n/2)(2a₁ + (n - 1)d)
Example: Sum of first 10 terms of 2, 5, 8, ... a₁ = 2, d = 3, n = 10 a₁₀ = 2 + 9(3) = 29 S₁₀ = (10/2)(2 + 29) = 5 × 31 = 155
Geometric Sequences
Each term is multiplied by a constant common ratio r.
n-th term: aₙ = a₁ · r^(n-1)
Example: 2, 6, 18, 54, ... a₁ = 2, r = 3 aₙ = 2 · 3^(n-1)
Sum of first n terms: Sₙ = a₁(1 - rⁿ)/(1 - r) for r ≠ 1
Example: Sum of first 5 terms of 3, 6, 12, ... a₁ = 3, r = 2, n = 5 S₅ = 3(1 - 2⁵)/(1 - 2) = 3(1 - 32)/(-1) = 3(-31)/(-1) = 93
Sum of first n terms when r = 1: Sₙ = n·a₁ (all terms are equal)
Infinite Geometric Series
Converges ONLY if |r| < 1.
Sum: S = a₁ / (1 - r)
Example: 1 + 1/2 + 1/4 + 1/8 + ... a₁ = 1, r = 1/2 S = 1 / (1 - 1/2) = 1 / (1/2) = 2
If |r| ≥ 1: The series diverges (no finite sum).
Sigma Notation
Σ (from i=1 to n) aᵢ = a₁ + a₂ + ... + aₙ
Example: Σ(i=1 to 4) i² = 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 = 30
Example: Σ(i=0 to 3) 2ⁱ = 2⁰ + 2¹ + 2² + 2³ = 1 + 2 + 4 + 8 = 15
Recursive Sequences
A recursive sequence defines each term using previous terms, not a direct formula.
Example (Fibonacci): F₁ = 1, F₂ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3.
F₃ = 1 + 1 = 2, F₄ = 1 + 2 = 3, F₅ = 2 + 3 = 5, F₆ = 3 + 5 = 8, ...
Example: a₁ = 3, aₙ = 2·aₙ₋₁ + 1 for n ≥ 2. a₂ = 2(3) + 1 = 7, a₃ = 2(7) + 1 = 15, a₄ = 2(15) + 1 = 31.
Key Terms
- 03 07 Sequences And Series
- Arithmetic Sequences
- Correct: B)
- Correct: C)
- Example 1: Arithmetic sequence
- Example 2: Geometric sequence
- Example 3: Infinite geometric series
- Geometric Sequences
- Infinite Geometric Series
- Recursive Sequences
- Sequences
- Sigma Notation
Worked Examples
Example 1: Arithmetic sequence
Find the 20th term and sum of first 20 terms of 5, 9, 13, ...
a₁ = 5, d = 4 a₂₀ = 5 + 19(4) = 5 + 76 = 81 S₂₀ = (20/2)(5 + 81) = 10 × 86 = 860
Example 2: Geometric sequence
Find the sum: 1 + 1/3 + 1/9 + ... + 1/3⁹
a₁ = 1, r = 1/3, n = 10 S₁₀ = 1(1 - (1/3)¹⁰)/(1 - 1/3) = (1 - 1/59049)/(2/3) = (59048/59049) × (3/2) = 88572/59049 ≈ 1.499
Example 3: Infinite geometric series
Sum: 4 + 2 + 1 + 1/2 + ...
a₁ = 4, r = 1/2 S = 4 / (1 - 1/2) = 4 / (1/2) = 8
Quiz
Q1: What does the concept of Arithmetic Sequences primarily refer to in this subject?
A) A visual representation of Arithmetic Sequences B) A computational error related to Arithmetic Sequences C) A historical anecdote about Arithmetic Sequences D) The definition and application of Arithmetic Sequences
Correct: D)
- If you chose A: This is incorrect. Arithmetic Sequences is defined as: the definition and application of arithmetic sequences. The other options describe different aspects that are not the primary focus.
- If you chose B: This is incorrect. Arithmetic Sequences is defined as: the definition and application of arithmetic sequences. The other options describe different aspects that are not the primary focus.
- If you chose C: This is incorrect. Arithmetic Sequences is defined as: the definition and application of arithmetic sequences. The other options describe different aspects that are not the primary focus.
- If you chose D: Arithmetic Sequences is defined as: the definition and application of arithmetic sequences. The other options describe different aspects that are not the primary focus. Correct!
Q2: What is the primary purpose of Geometric Sequences?
A) It is used only in advanced research contexts B) It is used to geometric sequences in mathematical analysis C) It replaces all other methods in this domain D) It is primarily a historical notation system
Correct: B)
- If you chose A: This is incorrect. Geometric Sequences serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose B: Geometric Sequences serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
- If you chose C: This is incorrect. Geometric Sequences serves the purpose described in the correct answer. The other options misrepresent its role.
- If you chose D: This is incorrect. Geometric Sequences serves the purpose described in the correct answer. The other options misrepresent its role.
Q3: Which statement about Infinite Geometric Series is TRUE?
A) Infinite Geometric Series is mentioned only as a historical footnote B) Infinite Geometric Series is an advanced topic beyond this subject's scope C) Infinite Geometric Series is not related to this subject D) Infinite Geometric Series is a fundamental concept covered in this subject
Correct: D)
- If you chose A: This is incorrect. Infinite Geometric Series is a fundamental concept covered in this subject. This subject covers Infinite Geometric Series as part of its core content.
- If you chose B: This is incorrect. Infinite Geometric Series is a fundamental concept covered in this subject. This subject covers Infinite Geometric Series as part of its core content.
- If you chose C: This is incorrect. Infinite Geometric Series is a fundamental concept covered in this subject. This subject covers Infinite Geometric Series as part of its core content.
- If you chose D: Infinite Geometric Series is a fundamental concept covered in this subject. This subject covers Infinite Geometric Series as part of its core content. Correct!
Q4: Based on the worked examples in this subject, what is the correct result?
A) 47. B) An unrelated numerical value C) A different result from a common mistake D) The inverse of the correct answer
Correct: A)
- If you chose A: The worked examples show that the result is 47.. The other options represent common errors. Correct!
- If you chose B: This is incorrect. The worked examples show that the result is 47.. The other options represent common errors.
- If you chose C: This is incorrect. The worked examples show that the result is 47.. The other options represent common errors.
- If you chose D: This is incorrect. The worked examples show that the result is 47.. The other options represent common errors.
Q5: How are Infinite Geometric Series and Recursive Sequences related?
A) Infinite Geometric Series is the inverse of Recursive Sequences B) Infinite Geometric Series and Recursive Sequences are completely unrelated topics C) Infinite Geometric Series and Recursive Sequences are closely related concepts D) Infinite Geometric Series is a special case of Recursive Sequences
Correct: C)
- If you chose A: This is incorrect. Both Infinite Geometric Series and Recursive Sequences are covered in this subject as interconnected topics.
- If you chose B: This is incorrect. Both Infinite Geometric Series and Recursive Sequences are covered in this subject as interconnected topics.
- If you chose C: Both Infinite Geometric Series and Recursive Sequences are covered in this subject as interconnected topics. Correct!
- If you chose D: This is incorrect. Both Infinite Geometric Series and Recursive Sequences are covered in this subject as interconnected topics.
Q6: What is a common pitfall when working with Sequences?
A) A common mistake is confusing Sequences with a similar concept B) The main error with Sequences is using it when it is not needed C) Sequences is always computed the same way in all contexts D) Sequences has no common misconceptions
Correct: A)
- If you chose A: Students often confuse Sequences with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
- If you chose B: This is incorrect. Students often confuse Sequences with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose C: This is incorrect. Students often confuse Sequences with similar-sounding or related concepts. Pay attention to the precise definitions.
- If you chose D: This is incorrect. Students often confuse Sequences with similar-sounding or related concepts. Pay attention to the precise definitions.
Q7: When should you apply Sigma Notation?
A) Apply Sigma Notation to solve problems in this subject's domain B) Avoid Sigma Notation unless explicitly instructed C) Sigma Notation is not practically useful D) Use Sigma Notation only in pure mathematics contexts
Correct: A)
- If you chose A: Sigma Notation is a practical tool used throughout this subject to solve relevant problems. Correct!
- If you chose B: This is incorrect. Sigma Notation is a practical tool used throughout this subject to solve relevant problems.
- If you chose C: This is incorrect. Sigma Notation is a practical tool used throughout this subject to solve relevant problems.
- If you chose D: This is incorrect. Sigma Notation is a practical tool used throughout this subject to solve relevant problems.
Practice Problems
-
10th term of arithmetic sequence: 2, 7, 12, ... Answer: a₁ = 2, d = 5. a₁₀ = 2 + 9(5) = 47.
-
Sum of first 8 terms of geometric sequence: 3, 6, 12, ... Answer: a₁ = 3, r = 2, n = 8. S₈ = 3(1 - 2⁸)/(1 - 2) = 3(1 - 256)/(-1) = 3(-255)/(-1) = 765.
-
Does 1 + 2 + 4 + 8 + ... converge? Answer: r = 2, |r| > 1. Diverges (no finite sum).
-
Sum of infinite series: 5 + 1 + 1/5 + ... Answer: a₁ = 5, r = 1/5. S = 5/(1 - 1/5) = 5/(4/5) = 25/4 = 6.25.
-
Evaluate Σ(i=1 to 5) (2i + 1) Answer: (2(1)+1) + (2(2)+1) + (2(3)+1) + (2(4)+1) + (2(5)+1) = 3 + 5 + 7 + 9 + 11 = 35.
-
Find the first 4 terms of the recursive sequence: a₁ = 2, aₙ = aₙ₋₁ + 3n for n ≥ 2. Answer: a₁ = 2. a₂ = 2 + 3(2) = 8. a₃ = 8 + 3(3) = 17. a₄ = 17 + 3(4) = 29.
Summary
Key takeaways:
- Arithmetic: aₙ = a₁ + (n-1)d, constant difference
- Geometric: aₙ = a₁·r^(n-1), constant ratio
- Arithmetic sum: Sₙ = (n/2)(a₁ + aₙ)
- Geometric sum: Sₙ = a₁(1-rⁿ)/(1-r)
- Infinite geometric converges only if |r| < 1: S = a₁/(1-r)
- Sigma notation Σ compactly represents sums
- Recursive sequences: defined by previous terms (e.g., Fibonacci)
Pitfalls
- Off-by-one errors in n-th term formulas: The arithmetic n-th term is a₁ + (n-1)d, not a₁ + n·d. The geometric n-th term is a₁·r^(n-1), not a₁·r^n. Using n instead of n-1 changes every result. Always verify by testing n = 1 — the formula should give a₁.
- Confusing arithmetic and geometric sum formulas: The arithmetic sum formula uses n/2 times the sum of first and last terms; the geometric sum formula uses a₁(1 - rⁿ)/(1 - r). Mixing these up (e.g., using the geometric formula for an arithmetic series) produces completely wrong answers.
- Forgetting the convergence condition for infinite geometric series: The infinite sum S = a₁/(1 - r) only works when |r| < 1. If |r| ≥ 1, the series diverges and has NO finite sum. Students often apply the formula blindly without checking r first.
- Misusing the geometric sum formula when r = 1: The formula Sₙ = a₁(1 - rⁿ)/(1 - r) involves division by zero when r = 1. In that case, every term equals a₁, so Sₙ = n·a₁. Always check for r = 1 before applying the standard formula.
- Misreading sigma notation indices: Σ(i=1 to n) starts at i = 1, but Σ(i=0 to n) includes the i = 0 term. Pay careful attention to the starting index, especially in geometric series where the 0th term differs (2⁰ = 1, not 2).
Next Steps
Next up: 03-08-mathematical-induction.md