Algebra Examples

$3$ , $2$ , $- 1$ , $- 6$ , $- 13$

Step 1

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

$- 1, - 3, - 5, - 7$

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 $- 2$ .

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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.

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

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

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Divide $- 2$ by $2$ .

$a = - 1$

Step 4

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

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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.

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Add $3$ to both sides of the equation.

$b = - 1 + 3$

Add $- 1$ and $3$ .

$b = 2$

Step 5

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $3$ .

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Set $a + b + c$ equal to the first term in the sequence $3$ .

$a + b + c = 3$

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

$- 1 + 2 + c = 3$

Add $- 1$ and $2$ .

$1 + c = 3$

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

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Subtract $1$ from both sides of the equation.

$c = 3 - 1$

Subtract $1$ from $3$ .

$c = 2$

Step 6

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

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