Extend Geometric Sequences Negatives and Fractions Calculator

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This calculator helps you extend geometric sequences that include negative numbers and fractions.

What is a 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 constant called the common ratio (r). The general form of a geometric sequence is:

a, ar, ar², ar³, ..., arⁿ⁻¹

Where:

a is the first term
r is the common ratio
n is the term number

Geometric sequences can include negative numbers and fractions, which adds complexity to their calculation and extension.

Formula for Extending Geometric Sequences

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

aₙ = a × r^(n-1)

Where:

aₙ is the nth term
a is the first term
r is the common ratio
n is the term number

This formula works for both positive and negative common ratios, as well as for sequences that include fractional terms.

Worked Examples

Example 1: Positive Common Ratio

Given a geometric sequence with first term a = 3 and common ratio r = 2, find the 5th term.

a₅ = 3 × 2^(5-1) = 3 × 16 = 48

The 5th term is 48.

Example 2: Negative Common Ratio

Given a geometric sequence with first term a = 5 and common ratio r = -3, find the 4th term.

a₄ = 5 × (-3)^(4-1) = 5 × 27 = 135

The 4th term is 135.

Example 3: Fractional Common Ratio

Given a geometric sequence with first term a = 4 and common ratio r = 1/2, find the 6th term.

a₆ = 4 × (1/2)^(6-1) = 4 × (1/32) = 1/8

The 6th term is 1/8.

FAQ

What is the difference between arithmetic and geometric sequences?

An arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms.

Can the common ratio be negative?

Yes, the common ratio can be negative, which means the sequence will alternate in sign.

How do I handle fractional common ratios?

Fractional common ratios are handled the same way as whole number ratios, using the formula aₙ = a × r^(n-1).

What if the first term is negative?

The first term can be negative, and the sequence will follow the same pattern based on the common ratio.