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.

General Form

A geometric sequence can be represented in general form as:

a, ar, ar^2, ar^3, ...

where:

• a = the first term
• r = the common ratio

nth Term Formula

The nth term of a geometric sequence can be found using the formula:

a_n = a * r^(n-1)

where:

• a_n = the nth term
• a = the first term
• r = the common ratio
• n = the term number

Sum of Terms Formula

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

S_n = a * (1 - rⁿ) / (1 - r)

where:

• S_n = sum of the first n terms
• a = the first term
• r = the common ratio
• n = the number of terms

Example

For example, consider the geometric sequence with a first term of 3 and a common ratio of 2. The first few terms of the sequence are 3, 6, 12, 24,...

To find the 5th term, we can use the nth term formula:

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

To find the sum of the first 4 terms, we can use the sum of terms formula:

S₄ = 3 * (1 - 2⁴) / (1 - 2) = 3 * (1 - 16) / (1 - 2) = 3 * (-15) / (-1) = 45