U1 Concept Review-Sequences & Series

Page 1: Sequences and Series

Key Concepts

  • Sequences

    • Ordered list of numbers determined by a mathematical pattern.

    • Examples: 1, 5, 9, 13, 17; 1000, 100, 10, 1.

    • n: Term position, natural number only.

  • Series

    • Sum of all terms in a finite sequence.

    • Examples: 5 + 10 + 15 + 20; 1 + 0.5 + 0.25 + 0.125.

Arithmetic Sequence

  • Has a common difference.

    • Example: 2, 4, 6, 8, 10 (d = 2).

Graph of an Arithmetic Sequence

  • Discrete graph; based on natural numbers.

  • Related to linear function:

    • y = mx + b (m = d).

    • Slope represents common difference.

    • t1 = b + m (first term value).

Arithmetic Series

  • Sum of an arithmetic sequence.

  • Formulas:

    • Known first term, last term, number of terms:tn = (n - 1)d + t1

  • First term, common difference, number of terms:S = n/2 (t1 + tn).

Page 2: Geometric Sequences

Geometric Sequence

  • Sequence with a common ratio.

    • Example: 3, 9, 27, 82, 243, 729 (r = 3).

  • Graph is discrete and not linear.

Finite Geometric Series

  • Sum of a finite geometric sequence.

  • Formulas:

    • Known values are: t1, r, n

    • Sum:S = t1 (1 - r^n) / (1 - r)

    • Alternate: S = tn (r - 1) / (r - 1).

Page 3: Infinite Geometric Series

Definition

  • Series without an end; can be convergent or divergent.

Convergence

  • Convergent:

    • If |r| < 1 (approaches finite value).

  • Divergent:

    • If |r| > 1 (sum grows infinitely).

Sum of Infinite Geometric Series

  • Only if convergent.

  • Formula: S = t1 / (1 - r) (for |r| < 1).

Vocabulary

  • Common Difference:

    • Difference in arithmetic sequence terms.

    • Formula: d = t(n) - t(n-1).

  • Common Ratio:

    • Ratio in geometric sequence terms.

    • Formula: r = t(n) / t(n-1).

  • Finite Sequence:

    • Sequence that ends with a final term.

  • Divergent vs. Convergent:

    • Divergent: |r| > 1

    • Convergent: |r| < 1.

Page 4: Common Errors

Description

  • Wrong Formulas:

    • Using general term instead of sum formula.

  • Confusing Sequences and Series:

    • Sequences list terms; series sum terms.

  • Divergent or Convergent:

    • Understand the definitions.

  • Discrete vs Continuous:

    • n represents natural number terms, leading to discrete graphs.