Geometric Sequence

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. The general form of a geometric sequence is:

a_n = a₁ * r^(n-1)

Where:

• a_n = the nth term of the sequence
• a₁ = the first term of the sequence
• r = the common ratio
• n = the term number

Finding the nth Term

To find the nth term of a geometric sequence, you can use the formula:

a_n = a₁ * r^(n-1)

Finding the Common Ratio

To find the common ratio of a geometric sequence, you can use the formula:

r = a₂ / a₁

Sum of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be found using the formula:

S_n = a₁ * (1 - rⁿ) / (1 - r)

Example

Let's consider a geometric sequence with a first term of 3 and a common ratio of 2. Find the 5th term and the sum of the first 5 terms.

Using the formula a_n = a₁ * r^(n-1), we find:

a₅ = 3 * 2^(5-1) = 3 * 2⁴ = 3 * 16 = 48

Using the formula S_n = a₁ * (1 - rⁿ) / (1 - r), we find:

S₅ = 3 * (1 - 2⁵) / (1 - 2) = 3 * (1 - 32) / (1 - 2) = 3 * (-31) / (-1) = 93

Practice Problems

1. Find the 8th term of the geometric sequence with a first term of 5 and a common ratio of 3.
2. Calculate the sum of the first 6 terms of the geometric sequence with a first term of 2 and a common ratio of 4.