Precalculus Examples

$2$ , $2 \sqrt{7}$ , $14$

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 $\sqrt{7}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \sqrt{7}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 2$ and $r = \sqrt{7}$ .

$a_{n} = 2 {\sqrt{7}}^{n - 1}$

Step 4

Substitute in the value of $n$ to find the $n$ th term.

$a_{4} = 2 {\sqrt{7}}^{(4) - 1}$

Step 5

Subtract $1$ from $4$ .

$a_{4} = 2 {\sqrt{7}}^{3}$

Step 6

Rewrite ${\sqrt{7}}^{3}$ as $\sqrt{7^{3}}$ .

$a_{4} = 2 \sqrt{7^{3}}$

Step 7

Raise $7$ to the power of $3$ .

$a_{4} = 2 \sqrt{343}$

Step 8

Rewrite $343$ as $7^{2} \cdot 7$ .

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Step 8.1

Factor $49$ out of $343$ .

$a_{4} = 2 \sqrt{49 (7)}$

Step 8.2

Rewrite $49$ as $7^{2}$ .

$a_{4} = 2 \sqrt{7^{2} \cdot 7}$

Step 9

Pull terms out from under the radical.

$a_{4} = 2 (7 \sqrt{7})$

Step 10

Multiply $7$ by $2$ .

$a_{4} = 14 \sqrt{7}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$a_{4} = 14 \sqrt{7}$

Decimal Form:

$a_{4} = 37.04051835 \dots$