Find the Next Term 2 , 5 , 11 , 20 , 32 , 47

$2$ , $5$ , $11$ , $20$ , $32$ , $47$

Step 1

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

$3, 6, 9, 12, 15$

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

$3$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $3$ .

Set $2 a$ equal to the constant second level difference $3$ .

$2 a = 3$

Divide each term in $2 a = 3$ by $2$ and simplify.

Divide each term in $2 a = 3$ by $2$ .

$\frac{2 a}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{3}{2}$

Divide $a$ by $1$ .

$a = \frac{3}{2}$

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 $\frac{3}{2}$ for $a$ .

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

Multiply $3 (\frac{3}{2})$ .

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

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

Multiply $3$ by $3$ .

$\frac{9}{2} + b = 3$

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

Subtract $\frac{9}{2}$ from both sides of the equation.

$b = 3 - \frac{9}{2}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$b = 3 \cdot \frac{2}{2} - \frac{9}{2}$

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

$b = \frac{3 \cdot 2}{2} - \frac{9}{2}$

Combine the numerators over the common denominator.

$b = \frac{3 \cdot 2 - 9}{2}$

Simplify the numerator.

Multiply $3$ by $2$ .

$b = \frac{6 - 9}{2}$

Subtract $9$ from $6$ .

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

Move the negative in front of the fraction.

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

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 $\frac{3}{2}$ for $a$ and $- \frac{3}{2}$ for $b$ .

$\frac{3}{2} - \frac{3}{2} + c = 2$

Simplify $\frac{3}{2} - \frac{3}{2} + c$ .

Combine the opposite terms in $\frac{3}{2} - \frac{3}{2} + c$ .

Combine the numerators over the common denominator.

$\frac{3 - 3}{2} + c = 2$

Subtract $3$ from $3$ .

$\frac{0}{2} + c = 2$

Divide $0$ by $2$ .

$0 + c = 2$

Add $0$ and $c$ .

$c = 2$

Step 6

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

$a_{n} = \frac{3}{2} n^{2} - \frac{3}{2} n + 2$

Step 7

Simplify each term.

Combine $\frac{3}{2}$ and $n^{2}$ .

$a_{n} = \frac{3 n^{2}}{2} - \frac{3}{2} n + 2$

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

$a_{n} = \frac{3 n^{2}}{2} - \frac{n \cdot 3}{2} + 2$

Move $3$ to the left of $n$ .

$a_{n} = \frac{3 n^{2}}{2} - \frac{3 n}{2} + 2$

Step 8

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

$a_{7} = \frac{3 {(7)}^{2}}{2} - \frac{3 (7)}{2} + 2$

Step 9

Combine the numerators over the common denominator.

$a_{7} = 2 + \frac{3 {(7)}^{2} - 3 \cdot 7}{2}$

Step 10

Simplify each term.

Raise $7$ to the power of $2$ .

$a_{7} = 2 + \frac{3 \cdot 49 - 3 \cdot 7}{2}$

Multiply $3$ by $49$ .

$a_{7} = 2 + \frac{147 - 3 \cdot 7}{2}$

Multiply $- 3$ by $7$ .

$a_{7} = 2 + \frac{147 - 21}{2}$

Step 11

Simplify the expression.

Subtract $21$ from $147$ .

$a_{7} = 2 + \frac{126}{2}$

Divide $126$ by $2$ .

$a_{7} = 2 + 63$

Add $2$ and $63$ .

$a_{7} = 65$

Find the Next Term 2 , 5 , 11 , 20 , 32 , 47

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