Algebra Examples

2

$2$ ,

6

$6$ ,

12

$12$ ,

20

$20$

Step 1

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

4, 6, 8

$4, 6, 8$

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

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

2

$2$

Step 3

Solve for

a

$a$ by setting

2 a

$2 a$ equal to the constant second level difference

2

$2$ .

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Step 3.1

Set

2 a

$2 a$ equal to the constant second level difference

2

$2$ .

2 a = 2

$2 a = 2$

Step 3.2

Divide each term in

2 a = 2

$2 a = 2$ by

2

$2$ and simplify.

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Step 3.2.1

Divide each term in

2 a = 2

$2 a = 2$ by

2

$2$ .

\frac{2 a}{2} = \frac{2}{2}

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

2

$2$ .

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Step 3.2.2.1.1

Cancel the common factor.

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

Step 3.2.2.1.2

Divide

$a$ by

$1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide

$2$ by

$2$ .

$a = 1$

Step 4

Solve for

$b$ by setting

$3 a + b$ equal to the first first level difference

$4$ .

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Step 4.1

Set

$3 a + b$ equal to the first first level difference

$4$ .

$3 a + b = 4$

Step 4.2

Substitute

$1$ for

$a$ .

$3 \cdot 1 + b = 4$

Step 4.3

Multiply

$3$ by

$1$ .

$3 + b = 4$

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

$3$ from both sides of the equation.

$b = 4 - 3$

Step 4.4.2

Subtract

$3$ from

$4$ .

$b = 1$

Step 5

Solve for

$c$ by setting

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

$2$ .

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Step 5.1

Set

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

$2$ .

$a + b + c = 2$

Step 5.2

Substitute

$1$ for

$a$ and

$1$ for

$b$ .

$1 + 1 + c = 2$

Step 5.3

Add

$1$ and

$1$ .

$2 + c = 2$

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

$2$ from both sides of the equation.

$c = 2 - 2$

Step 5.4.2

Subtract

$2$ from

$2$ .

$c = 0$

Step 6

Substitute the values of

$a$ ,

$b$ , and

$c$ into the quadratic sequence formula

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

$a_{n} = 1 n^{2} + 1 n + 0$

Step 7

Simplify.

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Step 7.1

Add

$1 n^{2} + 1 n$ and

$0$ .

$a_{n} = 1 n^{2} + 1 n$

Step 7.2

Simplify each term.

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Step 7.2.1

Multiply

$n^{2}$ by

$1$ .

$a_{n} = n^{2} + 1 n$

Step 7.2.2

Multiply

$n$ by

$1$ .

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