Basic Math Examples

$a n = 4 {(- \frac{3}{5})}^{n - 1}$

Step 1

Simplify $4 {(- \frac{3}{5})}^{n - 1}$ .

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Step 1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1.1

Apply the product rule to $- \frac{3}{5}$ .

$a n = 4 ({(- 1)}^{n - 1} {(\frac{3}{5})}^{n - 1})$

Step 1.1.2

Apply the product rule to $\frac{3}{5}$ .

$a n = 4 ({(- 1)}^{n - 1} \frac{3^{n - 1}}{5^{n - 1}})$

Step 1.2

Multiply ${(- 1)}^{n - 1} \frac{3^{n - 1}}{5^{n - 1}}$ .

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Step 1.2.1

Combine ${(- 1)}^{n - 1}$ and $\frac{3^{n - 1}}{5^{n - 1}}$ .

$a n = 4 \frac{{(- 1)}^{n - 1} \cdot 3^{n - 1}}{5^{n - 1}}$

Step 1.2.2

Rewrite ${(- 1)}^{n - 1} \cdot 3^{n - 1}$ as ${(- 1 \cdot 3)}^{n - 1}$ .

$a n = 4 \frac{{(- 1 \cdot 3)}^{n - 1}}{5^{n - 1}}$

Step 1.2.3

Multiply $- 1$ by $3$ .

$a n = 4 \frac{{(- 3)}^{n - 1}}{5^{n - 1}}$

Step 1.3

Combine $4$ and $\frac{{(- 3)}^{n - 1}}{5^{n - 1}}$ .

$a n = \frac{4 {(- 3)}^{n - 1}}{5^{n - 1}}$

Step 2

Since $n$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{4 {(- 3)}^{n - 1}}{5^{n - 1}} = a n$

Step 3

Move all terms containing $n$ to the left side of the equation.

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Step 3.1

Subtract $a n$ from both sides of the equation.

$\frac{4 {(- 3)}^{n - 1}}{5^{n - 1}} - a n = 0$

Step 3.2

To write $- a n$ as a fraction with a common denominator, multiply by $\frac{5^{n - 1}}{5^{n - 1}}$ .

$\frac{4 {(- 3)}^{n - 1}}{5^{n - 1}} - a n \cdot \frac{5^{n - 1}}{5^{n - 1}} = 0$

Step 3.3

Combine $- a n$ and $\frac{5^{n - 1}}{5^{n - 1}}$ .

$\frac{4 {(- 3)}^{n - 1}}{5^{n - 1}} + \frac{- a n \cdot 5^{n - 1}}{5^{n - 1}} = 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{4 {(- 3)}^{n - 1} - a n \cdot 5^{n - 1}}{5^{n - 1}} = 0$