Algebra Examples

$\frac{1}{4}$ , $\frac{2}{5}$ , $\frac{11}{20}$ , $\frac{7}{10}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $\frac{3}{20}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = \frac{3}{20}$

Step 2

This is the formula of an arithmetic sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{1}{4}$ and $d = \frac{3}{20}$ .

$a_{n} = \frac{1}{4} + \frac{3}{20} (n - 1)$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$a_{n} = \frac{1}{4} + \frac{3}{20} n + \frac{3}{20} \cdot - 1$

Step 4.2

Combine $\frac{3}{20}$ and $n$ .

$a_{n} = \frac{1}{4} + \frac{3 n}{20} + \frac{3}{20} \cdot - 1$

Step 4.3

Multiply $\frac{3}{20} \cdot - 1$ .

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Step 4.3.1

Combine $\frac{3}{20}$ and $- 1$ .

$a_{n} = \frac{1}{4} + \frac{3 n}{20} + \frac{3 \cdot - 1}{20}$

Step 4.3.2

Multiply $3$ by $- 1$ .

$a_{n} = \frac{1}{4} + \frac{3 n}{20} + \frac{- 3}{20}$

Step 4.4

Move the negative in front of the fraction.

$a_{n} = \frac{1}{4} + \frac{3 n}{20} - \frac{3}{20}$

Step 5

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a_{n} = \frac{3 n}{20} + \frac{1}{4} \cdot \frac{5}{5} - \frac{3}{20}$

Step 6

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{1}{4}$ by $\frac{5}{5}$ .

$a_{n} = \frac{3 n}{20} + \frac{5}{4 \cdot 5} - \frac{3}{20}$

Step 6.2

Multiply $4$ by $5$ .

$a_{n} = \frac{3 n}{20} + \frac{5}{20} - \frac{3}{20}$

Step 7

Combine the numerators over the common denominator.

$a_{n} = \frac{3 n}{20} + \frac{5 - 3}{20}$

Step 8

Subtract $3$ from $5$ .

$a_{n} = \frac{3 n}{20} + \frac{2}{20}$

Step 9

Cancel the common factor of $2$ and $20$ .

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Step 9.1

Factor $2$ out of $2$ .

$a_{n} = \frac{3 n}{20} + \frac{2 (1)}{20}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $2$ out of $20$ .

$a_{n} = \frac{3 n}{20} + \frac{2 \cdot 1}{2 \cdot 10}$

Step 9.2.2

Cancel the common factor.

$a_{n} = \frac{3 n}{20} + \frac{2 \cdot 1}{2 \cdot 10}$

Step 9.2.3

Rewrite the expression.

$a_{n} = \frac{3 n}{20} + \frac{1}{10}$

Step 10

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

$a_{5} = \frac{3 (5)}{20} + \frac{1}{10}$

Step 11

Multiply $3$ by $5$ .

$a_{5} = \frac{15}{20} + \frac{1}{10}$

Step 12

Cancel the common factor of $15$ and $20$ .

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Step 12.1

Factor $5$ out of $15$ .

$a_{5} = \frac{5 (3)}{20} + \frac{1}{10}$

Step 12.2

Cancel the common factors.

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Step 12.2.1

Factor $5$ out of $20$ .

$a_{5} = \frac{5 \cdot 3}{5 \cdot 4} + \frac{1}{10}$

Step 12.2.2

Cancel the common factor.

$a_{5} = \frac{5 \cdot 3}{5 \cdot 4} + \frac{1}{10}$

Step 12.2.3

Rewrite the expression.

$a_{5} = \frac{3}{4} + \frac{1}{10}$

Step 13

To write $\frac{3}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a_{5} = \frac{3}{4} \cdot \frac{5}{5} + \frac{1}{10}$

Step 14

To write $\frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a_{5} = \frac{3}{4} \cdot \frac{5}{5} + \frac{1}{10} \cdot \frac{2}{2}$

Step 15

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.1

Multiply $\frac{3}{4}$ by $\frac{5}{5}$ .

$a_{5} = \frac{3 \cdot 5}{4 \cdot 5} + \frac{1}{10} \cdot \frac{2}{2}$

Step 15.2

Multiply $4$ by $5$ .

$a_{5} = \frac{3 \cdot 5}{20} + \frac{1}{10} \cdot \frac{2}{2}$

Step 15.3

Multiply $\frac{1}{10}$ by $\frac{2}{2}$ .

$a_{5} = \frac{3 \cdot 5}{20} + \frac{2}{10 \cdot 2}$

Step 15.4

Multiply $10$ by $2$ .

$a_{5} = \frac{3 \cdot 5}{20} + \frac{2}{20}$

Step 16

Combine the numerators over the common denominator.

$a_{5} = \frac{3 \cdot 5 + 2}{20}$

Step 17

Simplify the numerator.

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Step 17.1

Multiply $3$ by $5$ .

$a_{5} = \frac{15 + 2}{20}$

Step 17.2

Add $15$ and $2$ .

$a_{5} = \frac{17}{20}$