Algebra Examples

$7$ , $9$ , $12$ , $16$ , $21$

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 $1$ .

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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.

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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.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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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 4

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

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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

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

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

Step 4.4

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

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Step 4.4.1

Subtract $\frac{3}{2}$ from 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.

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Step 4.4.5.1

Multiply $2$ by $2$ .

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

Step 4.4.5.2

Subtract $3$ from $4$ .

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

Step 5

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $7$ .

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Step 5.1

Set $a + b + c$ equal to the first term in the sequence $7$ .

$a + b + c = 7$

Step 5.2

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

$\frac{1}{2} + \frac{1}{2} + c = 7$

Step 5.3

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

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Step 5.3.1

Combine the numerators over the common denominator.

$c + \frac{1 + 1}{2} = 7$

Step 5.3.2

Simplify the expression.

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Step 5.3.2.1

Add $1$ and $1$ .

$c + \frac{2}{2} = 7$

Step 5.3.2.2

Divide $2$ by $2$ .

$c + 1 = 7$

Step 5.4

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

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Step 5.4.1

Subtract $1$ from both sides of the equation.

$c = 7 - 1$

Step 5.4.2

Subtract $1$ from $7$ .

$c = 6$

Step 6

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

$a_{n} = \frac{1}{2} n^{2} + \frac{1}{2} n + 6$

Step 7

Simplify each term.

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Step 7.1

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

$a_{n} = \frac{n^{2}}{2} + \frac{1}{2} n + 6$

Step 7.2

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

$a_{n} = \frac{n^{2}}{2} + \frac{n}{2} + 6$

Step 8

Substitute in the value of $n$ to find the $n$ th term.

$a_{6} = \frac{{(6)}^{2}}{2} + \frac{6}{2} + 6$

Step 9

Combine the numerators over the common denominator.

$a_{6} = 6 + \frac{{(6)}^{2} + 6}{2}$

Step 10

Simplify the expression.

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Step 10.1

Raise $6$ to the power of $2$ .

$a_{6} = 6 + \frac{36 + 6}{2}$

Step 10.2

Add $36$ and $6$ .

$a_{6} = 6 + \frac{42}{2}$

Step 10.3

Divide $42$ by $2$ .

$a_{6} = 6 + 21$

Step 10.4

Add $6$ and $21$ .

$a_{6} = 27$