Linear Algebra Examples

$a n = a r (n - 1)$

Step 1

Simplify $a r (n - 1)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$a n = a r n + a r \cdot - 1$

Step 1.1.2

Move $- 1$ to the left of $a r$ .

$a n = a r n - 1 \cdot (a r)$

Step 1.2

Rewrite $- 1 (a r)$ as $- (a r)$ .

$a n = a r n - a r$

Step 2

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

$a r n - a r = a n$

Step 3

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

$a r n - a r - a n = 0$

Step 4

Factor $a$ out of $a r n - a r - a n$ .

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Step 4.1

Factor $a$ out of $a r n$ .

$a (r n) - a r - a n = 0$

Step 4.2

Factor $a$ out of $- a r$ .

$a (r n) + a (- 1 r) - a n = 0$

Step 4.3

Factor $a$ out of $- a n$ .

$a (r n) + a (- 1 r) + a (- 1 n) = 0$

Step 4.4

Factor $a$ out of $a (r n) + a (- 1 r)$ .

$a (r n - 1 r) + a (- 1 n) = 0$

Step 4.5

Factor $a$ out of $a (r n - 1 r) + a (- 1 n)$ .

$a (r n - 1 r - 1 n) = 0$

Step 5

Rewrite $- 1 r$ as $- r$ .

$a (r n - r - 1 n) = 0$

Step 6

Rewrite $- 1 n$ as $- n$ .

$a (r n - r - n) = 0$

Step 7

Divide each term in $a (r n - r - n) = 0$ by $r n - r - n$ and simplify.

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Step 7.1

Divide each term in $a (r n - r - n) = 0$ by $r n - r - n$ .

$\frac{a (r n - r - n)}{r n - r - n} = \frac{0}{r n - r - n}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $r n - r - n$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{a (r n - r - n)}{r n - r - n} = \frac{0}{r n - r - n}$

Step 7.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{r n - r - n}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $r n - r - n$ .

$a = 0$

Step 8

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 9