Identify the Sequence 14 , 21 , 42 , 77

$14$ , $21$ , $42$ , $77$

Step 1

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

$7, 21, 35$

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

$14$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $14$ .

Set $2 a$ equal to the constant second level difference $14$ .

$2 a = 14$

Divide each term in $2 a = 14$ by $2$ and simplify.

Divide each term in $2 a = 14$ by $2$ .

$\frac{2 a}{2} = \frac{14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{14}{2}$

Divide $a$ by $1$ .

$a = \frac{14}{2}$

Simplify the right side.

Divide $14$ by $2$ .

$a = 7$

Step 4

Solve for $b$ by setting $3 a + b$ equal to the first first level difference $7$ .

Set $3 a + b$ equal to the first first level difference $7$ .

$3 a + b = 7$

Substitute $7$ for $a$ .

$3 \cdot 7 + b = 7$

Multiply $3$ by $7$ .

$21 + b = 7$

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

Subtract $21$ from both sides of the equation.

$b = 7 - 21$

Subtract $21$ from $7$ .

$b = - 14$

Step 5

Solve for $c$ by setting $a + b + c$ equal to the first term in the sequence $14$ .

Set $a + b + c$ equal to the first term in the sequence $14$ .

$a + b + c = 14$

Substitute $7$ for $a$ and $- 14$ for $b$ .

$7 - 14 + c = 14$

Subtract $14$ from $7$ .

$- 7 + c = 14$

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

Add $7$ to both sides of the equation.

$c = 14 + 7$

Add $14$ and $7$ .

$c = 21$

Step 6

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

$a_{n} = 7 n^{2} - 14 n + 21$

Identify the Sequence 14 , 21 , 42 , 77

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