Precalculus Examples

$5$ , $25$ , $125$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $5$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = 5$

Step 2

This is the form of a geometric sequence.

$a_{n} = a_{1} r^{n - 1}$

Step 3

Substitute in the values of $a_{1} = 5$ and $r = 5$ .

$a_{n} = 5 \cdot 5^{n - 1}$

Step 4

Multiply $5$ by $5^{n - 1}$ by adding the exponents.

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Step 4.1

Multiply $5$ by $5^{n - 1}$ .

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Step 4.1.1

Raise $5$ to the power of $1$ .

$a_{n} = 5^{1} \cdot 5^{n - 1}$

Step 4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a_{n} = 5^{1 + n - 1}$

Step 4.2

Combine the opposite terms in $1 + n - 1$ .

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Step 4.2.1

Subtract $1$ from $1$ .

$a_{n} = 5^{n + 0}$

Step 4.2.2

Add $n$ and $0$ .

$a_{n} = 5^{n}$