Arithmetic Sequences and Infinite Geometric Series

Arithmetic Sequence

• A sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.
• General form: a, a + d, a + 2d, a + 3d, ..., where a is the first term and d is the common difference.
• Formula to find the n-th term: a_n = a + (n - 1)d
• Formula to find the sum of the first n terms: S_n = (n/2)(2a + (n - 1)d)

Infinite Geometric Series

• A series of the form a + ar + ar^2 + ar^3 + ..., where a is the first term and r is the common ratio.
• Converges if |r| < 1, and diverges if |r| ≥ 1.
• Formula for the sum of an infinite geometric series: S = a / (1 - r), where |r| < 1.
• The sum of an infinite geometric series is finite and exists only if the common ratio is between -1 and 1.
• Applications of infinite geometric series include:
  • Modeling population growth and decay
  • Calculating interest rates and investments
  • Analyzing electronic circuits

Arithmetic Sequence

• An arithmetic sequence is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term.
• The general form of an arithmetic sequence is a, a + d, a + 2d, a + 3d,..., where a is the first term and d is the common difference.
• The formula to find the n-th term of an arithmetic sequence is a_n = a + (n - 1)d.
• The formula to find the sum of the first n terms of an arithmetic sequence is S_n = (n/2)(2a + (n - 1)d).

Infinite Geometric Series

• An infinite geometric series is a series of the form a + ar + ar^2 + ar^3 +..., where a is the first term and r is the common ratio.
• An infinite geometric series converges if |r| < 1, and diverges if |r| ≥ 1.
• The formula for the sum of an infinite geometric series is S = a / (1 - r), where |r| < 1.
• The sum of an infinite geometric series is finite and exists only if the common ratio is between -1 and 1.
• Infinite geometric series have applications in:
  • Modeling population growth and decay
  • Calculating interest rates and investments
  • Analyzing electronic circuits

Learn about arithmetic sequences and infinite geometric series, including their formulas and applications. Practice problems and quizzes to test your understanding.