1.3.3 Geometric Sequences and Series

Geometric Sequence

• In a geometric sequence, there is a common ratio ‘r’ between consecutive terms in the sequence

• A geometric sequence can be increasing (r > 1) or decreasing (0 < r < 1)

Common Ratio - r

The constant ratio between consecutive terms in a geometric sequence

• If the common ratio is a negative number the terms will alternate between positive and negative values

Geometric Sequence Formula - Finding the nth Term

Solving Geometric Sequence Problems

• Given a term and asked to find the first term or the common ratio

  • Substitute the information into the formula and solve the equation

• Given two or more consecutive terms and asked to find both the first term and the common ratio

  • Find the common ratio by dividing a term by the one before it

  • Substitute this and one of the terms into the formula to find the first term

• Given a term and the formula for the n^th term and asked to find the value of ‘n’

  • Solve these using logarithms as this type of problem sets up as an exponential equation

Geometric Series

A geometric series is the sum of a certain number of terms in a geometric sequence

Geometric Series Formula - Sum of the First ‘n’ Terms

• The left version is more convenient if r > 1

• The right version is more convenient if r < 1

Solving Geometric Series Problems

Given the sum of a certain number of terms and asked to find the value of the first term, the common ratio, or the number of terms within the sequence

• Substitute the information into the formula and solve the equation

Sum to Infinity

• A geometric sequence will either increase (positive numbers) or decrease (negative numbers) away from zero

• Or the terms will get progressively closer to zero

Convergence

• If the terms are getting closer to zero then the series is said to converge

  • This means that the sum of the series will approach a limiting value

  • Terms will get closer to zero if the common ratio ‘r’ is between 1 and -1

Signs of a Converging Sequence

Formula for a Converging Sequence Where |r| < 1