Find the Next Term 14 , 38 , 74 , 122 , 182 , 254

$14$ , $38$ , $74$ , $122$ , $182$ , $254$

Step 1

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

$24, 36, 48, 60, 72$

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

$12$

Step 3

Solve for $a$ by setting $2 a$ equal to the constant second level difference $12$ .

Set $2 a$ equal to the constant second level difference $12$ .

$2 a = 12$

Divide each term in $2 a = 12$ by $2$ and simplify.

Divide each term in $2 a = 12$ by $2$ .

$\frac{2 a}{2} = \frac{12}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{12}{2}$

Divide $a$ by $1$ .

$a = \frac{12}{2}$

Simplify the right side.

Divide $12$ by $2$ .

$a = 6$

Step 4

Solve for $b$ by setting $3 a + b$ equal to the first first level difference $24$ .

Set $3 a + b$ equal to the first first level difference $24$ .

$3 a + b = 24$

Substitute $6$ for $a$ .

$3 \cdot 6 + b = 24$

Multiply $3$ by $6$ .

$18 + b = 24$

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

Subtract $18$ from both sides of the equation.

$b = 24 - 18$

Subtract $18$ from $24$ .

$b = 6$

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 $6$ for $a$ and $6$ for $b$ .

$6 + 6 + c = 14$

Add $6$ and $6$ .

$12 + c = 14$

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

Subtract $12$ from both sides of the equation.

$c = 14 - 12$

Subtract $12$ from $14$ .

$c = 2$

Step 6

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

$a_{n} = 6 n^{2} + 6 n + 2$

Step 7

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

$a_{7} = 6 {(7)}^{2} + 6 (7) + 2$

Step 8

Simplify each term.

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

$a_{7} = 6 \cdot 49 + 6 (7) + 2$

Multiply $6$ by $49$ .

$a_{7} = 294 + 6 (7) + 2$

Multiply $6$ by $7$ .

$a_{7} = 294 + 42 + 2$

Step 9

Simplify by adding numbers.

Add $294$ and $42$ .

$a_{7} = 336 + 2$

Add $336$ and $2$ .

$a_{7} = 338$

Find the Next Term 14 , 38 , 74 , 122 , 182 , 254

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