Identify the Sequence 5 , 1 , 1 , 5

$5$ , $1$ , $1$ , $5$

Step 1

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

$- 4, 0, 4$

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

$4$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $4$ .

Set $2 a$ equal to the constant second level difference $4$ .

$2 a = 4$

Divide each term in $2 a = 4$ by $2$ and simplify.

Divide each term in $2 a = 4$ by $2$ .

$\frac{2 a}{2} = \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{4}{2}$

Divide $a$ by $1$ .

$a = \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$a = 2$

Step 4

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

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

$3 a + b = - 4$

Substitute $2$ for $a$ .

$3 \cdot 2 + b = - 4$

Multiply $3$ by $2$ .

$6 + b = - 4$

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

Subtract $6$ from both sides of the equation.

$b = - 4 - 6$

Subtract $6$ from $- 4$ .

$b = - 10$

Step 5

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 $2$ for $a$ and $- 10$ for $b$ .

$2 - 10 + c = 5$

Subtract $10$ from $2$ .

$- 8 + c = 5$

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

Add $8$ to both sides of the equation.

$c = 5 + 8$

Add $5$ and $8$ .

$c = 13$

Step 6

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

$a_{n} = 2 n^{2} - 10 n + 13$

Identify the Sequence 5 , 1 , 1 , 5

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