Precalculus Examples

$10$ , $8$ , $5$ , $1$ , $- 4$

Step 1

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

$- 2, - 3, - 4, - 5$

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

$- 1$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $- 1$ .

Tap for more steps...

Step 3.1

Set $2 a$ equal to the constant second level difference $- 1$ .

$2 a = - 1$

Step 3.2

Divide each term in $2 a = - 1$ by $2$ and simplify.

Tap for more steps...

Step 3.2.1

Divide each term in $2 a = - 1$ by $2$ .

$\frac{2 a}{2} = \frac{- 1}{2}$

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.

$\frac{2 a}{2} = \frac{- 1}{2}$

Step 3.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 1}{2}$

Step 3.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.3.1

Move the negative in front of the fraction.

$a = - \frac{1}{2}$

Step 4

Solve for $b$ by setting $3 a + b$ equal to the first first level difference $- 2$ .

Tap for more steps...

Step 4.1

Set $3 a + b$ equal to the first first level difference $- 2$ .

$3 a + b = - 2$

Step 4.2

Substitute $- \frac{1}{2}$ for $a$ .

$3 (- \frac{1}{2}) + b = - 2$

Step 4.3

Simplify each term.

Tap for more steps...

Step 4.3.1

Multiply $3 (- \frac{1}{2})$ .

Tap for more steps...

Step 4.3.1.1

Multiply $- 1$ by $3$ .

$- 3 (\frac{1}{2}) + b = - 2$

Step 4.3.1.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{- 3}{2} + b = - 2$

Step 4.3.2

Move the negative in front of the fraction.

$- \frac{3}{2} + b = - 2$

Step 4.4

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

Tap for more steps...

Step 4.4.1

Add $\frac{3}{2}$ to both sides of the equation.

$b = - 2 + \frac{3}{2}$

Step 4.4.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = - 2 \cdot \frac{2}{2} + \frac{3}{2}$

Step 4.4.3

Combine $- 2$ and $\frac{2}{2}$ .

$b = \frac{- 2 \cdot 2}{2} + \frac{3}{2}$

Step 4.4.4

Combine the numerators over the common denominator.

$b = \frac{- 2 \cdot 2 + 3}{2}$

Step 4.4.5

Simplify the numerator.

Tap for more steps...

Step 4.4.5.1

Multiply $- 2$ by $2$ .

$b = \frac{- 4 + 3}{2}$

Step 4.4.5.2

Add $- 4$ and $3$ .

$b = \frac{- 1}{2}$

Step 4.4.6

Move the negative in front of the fraction.

$b = - \frac{1}{2}$

Step 5

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $10$ .

Tap for more steps...

Step 5.1

Set $a + b + c$ equal to the first term in the sequence $10$ .

$a + b + c = 10$

Step 5.2

Substitute $- \frac{1}{2}$ for $a$ and $- \frac{1}{2}$ for $b$ .

$- \frac{1}{2} - \frac{1}{2} + c = 10$

Step 5.3

Simplify $- \frac{1}{2} - \frac{1}{2} + c$ .

Tap for more steps...

Step 5.3.1

Combine the numerators over the common denominator.

$c + \frac{- 1 - 1}{2} = 10$

Step 5.3.2

Simplify the expression.

Tap for more steps...

Step 5.3.2.1

Subtract $1$ from $- 1$ .

$c + \frac{- 2}{2} = 10$

Step 5.3.2.2

Divide $- 2$ by $2$ .

$c - 1 = 10$

Step 5.4

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

Tap for more steps...

Step 5.4.1

Add $1$ to both sides of the equation.

$c = 10 + 1$

Step 5.4.2

Add $10$ and $1$ .

$c = 11$

Step 6

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

$a_{n} = - \frac{1}{2} n^{2} - \frac{1}{2} n + 11$

Step 7

Simplify each term.

Tap for more steps...

Step 7.1

Combine $n^{2}$ and $\frac{1}{2}$ .

$a_{n} = - \frac{n^{2}}{2} - \frac{1}{2} n + 11$

Step 7.2

Combine $n$ and $\frac{1}{2}$ .

$a_{n} = - \frac{n^{2}}{2} - \frac{n}{2} + 11$