Identify the Sequence -21 , -9 , 11 , 39

$- 21$ , $- 9$ , $11$ , $39$

Step 1

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

$12, 20, 28$

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

$8$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $8$ .

Set $2 a$ equal to the constant second level difference $8$ .

$2 a = 8$

Divide each term in $2 a = 8$ by $2$ and simplify.

Divide each term in $2 a = 8$ by $2$ .

$\frac{2 a}{2} = \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{8}{2}$

Divide $a$ by $1$ .

$a = \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$a = 4$

Step 4

Solve for $b$ by setting $3 a + b$ equal to the first first level difference $12$ .

Set $3 a + b$ equal to the first first level difference $12$ .

$3 a + b = 12$

Substitute $4$ for $a$ .

$3 \cdot 4 + b = 12$

Multiply $3$ by $4$ .

$12 + b = 12$

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

Subtract $12$ from both sides of the equation.

$b = 12 - 12$

Subtract $12$ from $12$ .

$b = 0$

Step 5

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $- 21$ .

Set $a + b + c$ equal to the first term in the sequence $- 21$ .

$a + b + c = - 21$

Substitute $4$ for $a$ and $0$ for $b$ .

$4 + 0 + c = - 21$

Add $4$ and $0$ .

$4 + c = - 21$

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

Subtract $4$ from both sides of the equation.

$c = - 21 - 4$

Subtract $4$ from $- 21$ .

$c = - 25$

Step 6

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

$a_{n} = 4 n^{2} + 0 n - 25$

Step 7

Simplify.

Multiply $0$ by $n$ .

$a_{n} = 4 n^{2} + 0 - 25$

Add $4 n^{2}$ and $0$ .

$a_{n} = 4 n^{2} - 25$

Identify the Sequence -21 , -9 , 11 , 39

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