Geometric sequences and series

Geometric Sequences and Series

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Definition of a Geometric Sequence

A geometric sequence can be defined as follows:

• The first term is denoted as a.
• The common ratio is denoted as r.
• The nth term of a geometric sequence can be expressed as:

T_n = a imes r^{(n-1)}

Here, T_n is the nth term, a is the first term, and r is the common ratio.

Example of a Geometric Sequence

Consider the sequence 2, 6, 18, 54, ...

In this sequence:

• The first term a = 2.
• The common ratio r = 6/2 = 3.

The sequence can be described using the formula:

T_n = 2 imes 3^{(n-1)}

To find the 5th term:

T_5 = 2 imes 3^{(5-1)} = 2 imes 3^4 = 2 imes 81 = 162

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. The sum of the first n terms of a geometric series can be calculated using the formula:

S_n = a imes rac{1 - r^n}{1 - r} ext{ (if } r eq 1 ext{)}

Where:

• S_n is the sum of the first n terms.
• a is the first term.
• r is the common ratio.

Example of a Geometric Series

Using the previous sequence 2, 6, 18, 54, ..., let's find the sum of the first 4 terms.

Here, a = 2, r = 3, and n = 4.

Using the formula:

S_4 = 2 imes rac{1 - 3^4}{1 - 3}

Calculating:

S_4 = 2 imes rac{1 - 81}{1 - 3} = 2 imes rac{-80}{-2} = 2 imes 40 = 80

Sum of an Infinite Geometric Series

If the absolute value of the common ratio is less than 1 (|r| < 1), the geometric series converges to a finite sum. The formula for the sum of an infinite geometric series is:

S = rac{a}{1 - r} ext{ (if } |r| < 1 ext{)}

Example of an Infinite Geometric Series

Consider the series 1, 1/2, 1/4, 1/8, ...

In this series:

• a = 1
• r = 1/2

Using the formula for the sum of an infinite series:

S = rac{1}{1 - 1/2} = rac{1}{1/2} = 2

Common Mistakes

Applications of Geometric Sequences and Series

Geometric sequences and series are used in various fields such as finance, physics, and computer science. For example, they can represent exponential growth or decay, such as population growth or radioactive decay.

Practice Problems

1. Find the 6th term of the geometric sequence where the first term is 5 and the common ratio is 2.
2. Calculate the sum of the first 5 terms of the geometric series with first term 3 and common ratio 4.
3. Determine the sum of the infinite geometric series with first term 7 and common ratio 1/3.
4. Explain why a geometric series with a common ratio greater than or equal to 1 does not converge.

Check your understanding

1. What is the formula for the nth term of a geometric sequence?
2. How do you calculate the sum of the first n terms of a geometric series?
3. What condition must be met for an infinite geometric series to converge?
4. Provide an example of a real-world situation that can be modelled using a geometric sequence.