Find the Next Term 4 , 16 , 36 , 64 , 100

$4$ , $16$ , $36$ , $64$ , $100$

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

$12, 20, 28, 36$

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

$8$

Solve for $a$ by setting $2 a$ equal to the constant second level difference $8$ .

Set $2 a$ equal to the constant second level difference $8$ .

$2 a = 8$

Divide each term in $2 a = 8$ by $2$ and simplify.

Divide each term in $2 a = 8$ by $2$ .

$\frac{2 a}{2} = \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{8}{2}$

Divide $a$ by $1$ .

$a = \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$a = 4$

Solve for $b$ by setting $3 a + b$ equal to the first first level difference $12$ .

Set $3 a + b$ equal to the first first level difference $12$ .

$3 a + b = 12$

Substitute $4$ for $a$ .

$3 \cdot 4 + b = 12$

Multiply $3$ by $4$ .

$12 + b = 12$

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

Subtract $12$ from both sides of the equation.

$b = 12 - 12$

Subtract $12$ from $12$ .

$b = 0$

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $4$ .

Set $a + b + c$ equal to the first term in the sequence $4$ .

$a + b + c = 4$

Substitute $4$ for $a$ and $0$ for $b$ .

$4 + 0 + c = 4$

Add $4$ and $0$ .

$4 + c = 4$

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

Subtract $4$ from both sides of the equation.

$c = 4 - 4$

Subtract $4$ from $4$ .

$c = 0$

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

$a_{n} = 4 n^{2} + 0 n + 0$

Simplify.

Add $4 n^{2} + 0 n$ and $0$ .

$a_{n} = 4 n^{2} + 0 n$

Multiply $0$ by $n$ .

$a_{n} = 4 n^{2} + 0$

Add $4 n^{2}$ and $0$ .

$a_{n} = 4 n^{2}$

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

$a_{6} = 4 {(6)}^{2}$

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

$a_{6} = 4 \cdot 36$

Multiply $4$ by $36$ .

$a_{6} = 144$

Find the Next Term 4 , 16 , 36 , 64 , 100

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