Find the Next Term -2 , 1 , 10 , 25 , 46

$- 2$ , $1$ , $10$ , $25$ , $46$

Step 1

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

$3, 9, 15, 21$

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

$6$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $6$ .

Set $2 a$ equal to the constant second level difference $6$ .

$2 a = 6$

Divide each term in $2 a = 6$ by $2$ and simplify.

Divide each term in $2 a = 6$ by $2$ .

$\frac{2 a}{2} = \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{6}{2}$

Divide $a$ by $1$ .

$a = \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$a = 3$

Step 4

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

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

$3 a + b = 3$

Substitute $3$ for $a$ .

$3 \cdot 3 + b = 3$

Multiply $3$ by $3$ .

$9 + b = 3$

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

Subtract $9$ from both sides of the equation.

$b = 3 - 9$

Subtract $9$ from $3$ .

$b = - 6$

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 $3$ for $a$ and $- 6$ for $b$ .

$3 - 6 + c = - 2$

Subtract $6$ from $3$ .

$- 3 + c = - 2$

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

Add $3$ to both sides of the equation.

$c = - 2 + 3$

Add $- 2$ and $3$ .

$c = 1$

Step 6

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

$a_{n} = 3 n^{2} - 6 n + 1$

Step 7

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

$a_{6} = 3 {(6)}^{2} - 6 \cdot 6 + 1$

Step 8

Simplify each term.

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

$a_{6} = 3 \cdot 36 - 6 \cdot 6 + 1$

Multiply $3$ by $36$ .

$a_{6} = 108 - 6 \cdot 6 + 1$

Multiply $- 6$ by $6$ .

$a_{6} = 108 - 36 + 1$

Step 9

Simplify by adding and subtracting.

Subtract $36$ from $108$ .

$a_{6} = 72 + 1$

Add $72$ and $1$ .

$a_{6} = 73$

Find the Next Term -2 , 1 , 10 , 25 , 46

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