Find the Next Term 2 , 8 , 18 , 32 , 50

$2$ , $8$ , $18$ , $32$ , $50$

Step 1

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

$6, 10, 14, 18$

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 $6$ .

Set $3 a + b$ equal to the first first level difference $6$ .

$3 a + b = 6$

Substitute $2$ for $a$ .

$3 \cdot 2 + b = 6$

Multiply $3$ by $2$ .

$6 + b = 6$

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

Subtract $6$ from both sides of the equation.

$b = 6 - 6$

Subtract $6$ from $6$ .

$b = 0$

Step 5

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $2$ .

Set $a + b + c$ equal to the first term in the sequence $2$ .

$a + b + c = 2$

Substitute $2$ for $a$ and $0$ for $b$ .

$2 + 0 + c = 2$

Add $2$ and $0$ .

$2 + c = 2$

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

Subtract $2$ from both sides of the equation.

$c = 2 - 2$

Subtract $2$ from $2$ .

$c = 0$

Step 6

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

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

Step 7

Simplify.

Add $2 n^{2} + 0 n$ and $0$ .

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

Multiply $0$ by $n$ .

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

Add $2 n^{2}$ and $0$ .

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

Step 8

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

$a_{6} = 2 {(6)}^{2}$

Step 9

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

$a_{6} = 2 \cdot 36$

Step 10

Multiply $2$ by $36$ .

$a_{6} = 72$

Find the Next Term 2 , 8 , 18 , 32 , 50

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