Precalculus Examples

$\frac{63}{4}$ , $\frac{1702}{32}$

Step 1

Cancel the common factor of $1702$ and $32$ .

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Step 1.1

Factor $2$ out of $1702$ .

$\frac{63}{4}, \frac{2 (851)}{32}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $2$ out of $32$ .

$\frac{63}{4}, \frac{2 \cdot 851}{2 \cdot 16}$

Step 1.2.2

Cancel the common factor.

$\frac{63}{4}, \frac{2 \cdot 851}{2 \cdot 16}$

Step 1.2.3

Rewrite the expression.

$\frac{63}{4}, \frac{851}{16}$

Step 2

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $\frac{599}{16}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = \frac{599}{16}$

Step 3

This is the formula of an arithmetic sequence.

$a_{n} = a_{1} + d (n - 1)$

Step 4

Substitute in the values of $a_{1} = \frac{63}{4}$ and $d = \frac{599}{16}$ .

$a_{n} = \frac{63}{4} + \frac{599}{16} (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$a_{n} = \frac{63}{4} + \frac{599}{16} n + \frac{599}{16} \cdot - 1$

Step 5.2

Combine $\frac{599}{16}$ and $n$ .

$a_{n} = \frac{63}{4} + \frac{599 n}{16} + \frac{599}{16} \cdot - 1$

Step 5.3

Multiply $\frac{599}{16} \cdot - 1$ .

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Step 5.3.1

Combine $\frac{599}{16}$ and $- 1$ .

$a_{n} = \frac{63}{4} + \frac{599 n}{16} + \frac{599 \cdot - 1}{16}$

Step 5.3.2

Multiply $599$ by $- 1$ .

$a_{n} = \frac{63}{4} + \frac{599 n}{16} + \frac{- 599}{16}$

Step 5.4

Move the negative in front of the fraction.

$a_{n} = \frac{63}{4} + \frac{599 n}{16} - \frac{599}{16}$

Step 6

To write $\frac{63}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a_{n} = \frac{599 n}{16} + \frac{63}{4} \cdot \frac{4}{4} - \frac{599}{16}$

Step 7

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

Multiply $\frac{63}{4}$ by $\frac{4}{4}$ .

$a_{n} = \frac{599 n}{16} + \frac{63 \cdot 4}{4 \cdot 4} - \frac{599}{16}$

Step 7.2

Multiply $4$ by $4$ .

$a_{n} = \frac{599 n}{16} + \frac{63 \cdot 4}{16} - \frac{599}{16}$

Step 8

Combine the numerators over the common denominator.

$a_{n} = \frac{599 n}{16} + \frac{63 \cdot 4 - 599}{16}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $63$ by $4$ .

$a_{n} = \frac{599 n}{16} + \frac{252 - 599}{16}$

Step 9.2

Subtract $599$ from $252$ .

$a_{n} = \frac{599 n}{16} + \frac{- 347}{16}$

Step 10

Move the negative in front of the fraction.

$a_{n} = \frac{599 n}{16} - \frac{347}{16}$

Step 11

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

$a_{3} = \frac{599 (3)}{16} - \frac{347}{16}$

Step 12

Combine the numerators over the common denominator.

$a_{3} = \frac{599 (3) - 347}{16}$

Step 13

Simplify the expression.

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Step 13.1

Multiply $599$ by $3$ .

$a_{3} = \frac{1797 - 347}{16}$

Step 13.2

Subtract $347$ from $1797$ .

$a_{3} = \frac{1450}{16}$

Step 14

Cancel the common factor of $1450$ and $16$ .

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Step 14.1

Factor $2$ out of $1450$ .

$a_{3} = \frac{2 (725)}{16}$

Step 14.2

Cancel the common factors.

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Step 14.2.1

Factor $2$ out of $16$ .

$a_{3} = \frac{2 \cdot 725}{2 \cdot 8}$

Step 14.2.2

Cancel the common factor.

$a_{3} = \frac{2 \cdot 725}{2 \cdot 8}$

Step 14.2.3

Rewrite the expression.

$a_{3} = \frac{725}{8}$