Overview

This topic introduces sequences and series, including how to identify, write, and evaluate arithmetic and geometric progressions. You'll also learn to use sigma notation and apply the binomial theorem to expand expressions raised to powers.

Key Concepts and Structures

• Sequence: An ordered list of numbers. Can be arithmetic, geometric, or neither.
• Arithmetic Sequence: Each term is generated by adding a constant difference (d).
  Formula: a_n = a_1 + (n - 1)d
• Geometric Sequence: Each term is generated by multiplying by a constant ratio (r).
  Formula: a_n = a_1 * r^{n-1}
• Series: The sum of the terms of a sequence.
• Arithmetic Series: S_n = n/2 (a_1 + a_n)
• Geometric Series:
  • S_n = a_1(1 - r^n) / (1 - r), if r ≠ 1
  • S = a_1 / (1 - r), for infinite series where |r| < 1
• Sigma Notation (∑): Compact form of expressing a series.
  Example: ∑_{i=1}^{4} i = 1 + 2 + 3 + 4
• Binomial Theorem: Expands expressions like (a + b)^n using combinations:
  (a + b)^n = ∑_{k=0}^{n} C(n,k)a^{n-k}b^k
• Combinations: C(n, k) = n! / (k!(n - k)!)

Step-by-Step Example

Problem: Find the 5th term of the arithmetic sequence where a_1 = 7 and d = 3

Step 1: Use the formula:
a_n = a_1 + (n - 1)d

Step 2: Substitute values:
a_5 = 7 + (5 - 1) * 3 = 7 + 12 = 19

Final Answer: a_5 = 19

Quick Tip