Calculus Examples

$3$ , $9$ , $27$

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 $3$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = 3$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 3$ and $r = 3$ .

$a_{n} = 3 \cdot 3^{n - 1}$

Step 4

Multiply $3$ by $3^{n - 1}$ by adding the exponents.

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Step 4.1

Multiply $3$ by $3^{n - 1}$ .

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Step 4.1.1

Raise $3$ to the power of $1$ .

$a_{n} = 3^{1} \cdot 3^{n - 1}$

Step 4.1.2

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

$a_{n} = 3^{1 + n - 1}$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from $1$ .

$a_{n} = 3^{n + 0}$

Step 4.2.2

Add $n$ and $0$ .

$a_{n} = 3^{n}$

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