8-5 Geometric Sequences

Property

In a geometric sequence, each term is multiplied by the same number to find the next term. Terms in a geometric sequence are not equally spaced on a number line.

Examples

• The sequence $1, 2, 4, 8, 16, ...$ is geometric because you multiply by 2 each time to get the next term.
• In the sequence $100, 50, 25, ...$, the rule is to multiply by $\frac{1}{2}$, so the next term is $12.5$.
• The sequence $1, 10, 100, ...$ is geometric. The rule is 'multiply by 10', so the next term is $100 \cdot 10 = 1000$.

Explanation

If an arithmetic sequence is like walking, a geometric sequence is like a rocket launch! Instead of adding, you multiply by the same number each time, causing the terms to grow (or shrink) incredibly fast. It’s all about multiplication, making for an explosive pattern where numbers can get huge or tiny in just a few steps.

Property

A sequence $a_1, a_2, a_3, \ldots$ is geometric if and only if every consecutive ratio is equal to the same nonzero constant $r$:

Property

The explicit formula for a geometric sequence is used to find any term in the sequence directly. The formula is given by:

where $a_n$ is the $n$-th term, $a_1$ is the first term, and $r$ is the common ratio.

Examples

• To write the explicit formula for a sequence with a first term of $3$ and a common ratio of $2$, we substitute $a_1 = 3$ and $r = 2$ into the formula: $a_n = 3 \cdot 2^{n-1}$.
• For the sequence $5, 15, 45, \dots$, the first term is $a_1 = 5$. The common ratio is $r = \frac{15}{5} = 3$. The explicit formula is $a_n = 5 \cdot 3^{n-1}$.
• For the sequence $100, 50, 25, \dots$, the first term is $a_1 = 100$ and the common ratio is $r = \frac{50}{100} = \frac{1}{2}$. The explicit formula is $a_n = 100 \cdot (\frac{1}{2})^{n-1}$.

Explanation

The explicit formula, also known as the general term, allows you to calculate any term in a geometric sequence without having to find all the preceding terms. It is defined using the sequence''s first term ($a_1$) and its common ratio ($r$). To write the formula, you simply substitute the known values of $a_1$ and $r$ into the standard equation $a_n = a_1 \cdot r^{n-1}$. This formula is a type of exponential function, where the term number $n$ acts as the variable in the exponent.