Lesson 4: Geometric Sequences

Property

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, $\frac{a_n}{a_{n-1}}$, is $r$, the common ratio. $n$ is greater than or equal to two.

Examples

• The sequence $5, 15, 45, 135, \ldots$ is geometric because the ratio between consecutive terms is always 3. The common ratio is $r=3$.

Property

A recursive formula for a geometric sequence expresses each term in relation to the previous term:

Examples

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.