Arithmetic and Geometric Series

Arithmetic and Geometric Series

Arithmetic Series Basics

  • An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This is also known as the ‘common difference’.
  • The general formula for an arithmetic sequence is a + (n - 1)d where ‘a’ is the first term, ‘n’ is the term number and ‘d’ is the common difference.
  • An arithmetic series can be represented as: sum = n/2 (2a + (n - 1)d), where ‘sum’ is the sum of the arithmetic series, ‘n’ is the number of terms, ‘a’ is the first term and ‘d’ is the common difference.

Arithmetic Series Properties

  • Commutativity: The sum of an arithmetic series remains the same regardless of the order in which the terms are added.
  • Associativity: Grouping the terms of an arithmetic series in different ways does not alter the sum.
  • You can find any term of an arithmetic series using the formula a + (n - 1)d.

Geometric Series Basics

  • A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ‘common ratio’.
  • The general formula for a geometric sequence is ar^(n-1) where ‘a’ is the first term, ‘r’ is the common ratio and ‘n’ is the term number.
  • A geometric series can be represented as: sum = a (1 - r^n) / (1 - r) for r < 1 where ‘a’ is the first term, ‘r’ is the common ratio, ‘n’ is the number of terms and ‘sum’ is the sum of the geometric series.

Geometric Series Properties

  • The sum of a geometric series depends on the common ratio ‘r’. If the absolute value of ‘r’ is less than 1, the series converges to a sum.
  • For a converging geometric series, the sum can be found using the formula sum = a /(1 - r).
  • Any term of a geometric series can be found using the formula ar^(n-1).

Strategies for Solving Problems

  • Use the correct formula: Make sure to use the correct formula for an arithmetic or geometric series based on whether the series has a constant difference or a constant ratio.
  • Identify the sequence : Identify whether the series is arithmetic or geometric by looking at the relationship between the terms.
  • Substitute the values : Substitute the given values into the formula to solve the problem.