Identify the Sequence -5 , -4 , -1 , 4 , 11 , 20 , 31

$- 5$ , $- 4$ , $- 1$ , $4$ , $11$ , $20$ , $31$

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

$1, 3, 5, 7, 9, 11$

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 $a_{n} = a n^{2} + b n + c$ .

$2$

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

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

$2 a = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Divide $2$ by $2$ .

$a = 1$

Solve for $b$ by setting $3 a + b$ equal to the first first level difference $1$ .

Set $3 a + b$ equal to the first first level difference $1$ .

$3 a + b = 1$

Substitute $1$ for $a$ .

$3 \cdot 1 + b = 1$

Multiply $3$ by $1$ .

$3 + b = 1$

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

Subtract $3$ from both sides of the equation.

$b = 1 - 3$

Subtract $3$ from $1$ .

$b = - 2$

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $- 5$ .

Set $a + b + c$ equal to the first term in the sequence $- 5$ .

$a + b + c = - 5$

Substitute $1$ for $a$ and $- 2$ for $b$ .

$1 - 2 + c = - 5$

Subtract $2$ from $1$ .

$- 1 + c = - 5$

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

Add $1$ to both sides of the equation.

$c = - 5 + 1$

Add $- 5$ and $1$ .

$c = - 4$

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

$a_{n} = 1 n^{2} - 2 n - 4$

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

$a_{n} = n^{2} - 2 n - 4$

Identify the Sequence -5 , -4 , -1 , 4 , 11 , 20 , 31

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