Algebra Examples

$5$ , $5 \sqrt{2}$ , $10$

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 $\sqrt{2}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \sqrt{2}$

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 $r = \sqrt{2}$ .

$a_{n} = 5 {\sqrt{2}}^{n - 1}$

Step 4

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

$a_{5} = 5 {\sqrt{2}}^{(5) - 1}$

Step 5

Subtract $1$ from $5$ .

$a_{5} = 5 {\sqrt{2}}^{4}$

Step 6

Rewrite ${\sqrt{2}}^{4}$ as $2^{2}$ .

Tap for more steps...

Step 6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$a_{5} = 5 {(2^{\frac{1}{2}})}^{4}$

Step 6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$a_{5} = 5 \cdot 2^{\frac{1}{2} \cdot 4}$

Step 6.3

Combine $\frac{1}{2}$ and $4$ .

$a_{5} = 5 \cdot 2^{\frac{4}{2}}$

Step 6.4

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 6.4.1

Factor $2$ out of $4$ .

$a_{5} = 5 \cdot 2^{\frac{2 \cdot 2}{2}}$

Step 6.4.2

Cancel the common factors.

Tap for more steps...

Step 6.4.2.1

Factor $2$ out of $2$ .

$a_{5} = 5 \cdot 2^{\frac{2 \cdot 2}{2 (1)}}$

Step 6.4.2.2

Cancel the common factor.

$a_{5} = 5 \cdot 2^{\frac{2 \cdot 2}{2 \cdot 1}}$

Step 6.4.2.3

Rewrite the expression.

$a_{5} = 5 \cdot 2^{\frac{2}{1}}$

Step 6.4.2.4

Divide $2$ by $1$ .

$a_{5} = 5 \cdot 2^{2}$

Step 7

Simplify the expression.

Tap for more steps...

Step 7.1

Raise $2$ to the power of $2$ .

$a_{5} = 5 \cdot 4$

Step 7.2

Multiply $5$ by $4$ .

$a_{5} = 20$