Algebra Examples

$2$ , $6$ , $12$ , $20$

Step 1

Find the first level differences by finding the differences between consecutive terms.

$4, 6, 8$

Step 2

Find the second level difference by finding the differences between the first level differences. Because the second level difference is constant, the sequence is quadratic and given by $a_{n} = a n^{2} + b n + c$ .

$2$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $2$ .

Tap for more steps...

Step 3.1

Set $2 a$ equal to the constant second level difference $2$ .

$2 a = 2$

Step 3.2

Divide each term in $2 a = 2$ by $2$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $2 a = 2$ by $2$ .

$\frac{2 a}{2} = \frac{2}{2}$

Step 3.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 a}{2} = \frac{2}{2}$

Step 3.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{2}{2}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Divide $2$ by $2$ .

$a = 1$

Step 4

Solve for $b$ by setting $3 a + b$ equal to the first first level difference $4$ .

Tap for more steps...

Step 4.1

Set $3 a + b$ equal to the first first level difference $4$ .

$3 a + b = 4$

Step 4.2

Substitute $1$ for $a$ .

$3 \cdot 1 + b = 4$

Step 4.3

Multiply $3$ by $1$ .

$3 + b = 4$

Step 4.4

Move all terms not containing $b$ to the right side of the equation.

Tap for more steps...

Step 4.4.1

Subtract $3$ from both sides of the equation.

$b = 4 - 3$

Step 4.4.2

Subtract $3$ from $4$ .

$b = 1$

Step 5

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $2$ .

Tap for more steps...

Step 5.1

Set $a + b + c$ equal to the first term in the sequence $2$ .

$a + b + c = 2$

Step 5.2

Substitute $1$ for $a$ and $1$ for $b$ .

$1 + 1 + c = 2$

Step 5.3

Add $1$ and $1$ .

$2 + c = 2$

Step 5.4

Move all terms not containing $c$ to the right side of the equation.

Tap for more steps...

Step 5.4.1

Subtract $2$ from both sides of the equation.

$c = 2 - 2$

Step 5.4.2

Subtract $2$ from $2$ .

$c = 0$

Step 6

Substitute the values of $a$ , $b$ , and $c$ into the quadratic sequence formula $a_{n} = a n^{2} + b n + c$ .

$a_{n} = 1 n^{2} + 1 n + 0$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Add $1 n^{2} + 1 n$ and $0$ .

$a_{n} = 1 n^{2} + 1 n$

Step 7.2

Simplify each term.

Tap for more steps...

Step 7.2.1

Multiply $n^{2}$ by $1$ .

$a_{n} = n^{2} + 1 n$

Step 7.2.2

Multiply $n$ by $1$ .

$a_{n} = n^{2} + n$