Write the explicit formula for a geometric sequence

A geometric sequence is determined by its first term $a_1$ and constant common ratio $r$ , so its $n$ th term is represented explicitly as $a_n=a_1r^{\,n-1}$ for positive integer indices. The formula expresses repeated multiplication, including growth, decay, alternating signs, and special cases such as $r=0$ , and distinguishes multiplicative change from the constant addition of an arithmetic sequence. This understanding supports exponential functions and finite geometric sums, but not infinite-series convergence or more abstract indexing.

Detailed Explanation: Write the explicit formula for a geometric sequence

A geometric sequence is a sequence in which each term is found by multiplying the previous term by the same constant number. This constant is called the common ratio, $r$ .

The explicit formula is

a_n=a_1r^{n-1},

where:

$a_1$ is the first term,
$r$ is the common ratio,
$n$ is the term number.

Worked example

Write an explicit formula for the sequence

5,\ -10,\ 20,\ -40,\ldots

Step 1: Identify the first term.

The first term is

a_1=5.

Step 2: Find the common ratio.

Divide a term by the term before it:

r=\frac{-10}{5}=-2.

Check the next pair:

\frac{20}{-10}=-2,

so the common ratio is constant.

Step 3: Substitute into the formula.

Using $a_n=a_1r^{n-1}$ :

a_n=5(-2)^{n-1}.

Therefore, the explicit formula is

\boxed{a_n=5(-2)^{n-1}}.

Step 4: Check the formula.

For $n=1$ :

a_1=5(-2)^0=5.

For $n=3$ :

a_3=5(-2)^2=5(4)=20.

These match the sequence.

The exponent $n-1$ shows how many times the common ratio is multiplied. A negative ratio causes the signs to alternate. In contrast, an arithmetic sequence uses repeated addition or subtraction, not repeated multiplication.

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Patterning - Equation from Rule for Geometric Pattern

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