Algebra Examples

$\frac{9}{10}$ , $\frac{3}{4}$ , $\frac{3}{5}$ , $\frac{9}{20}$

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 $a_{1} = \frac{9}{10}$ and $d = - \frac{3}{20}$ .

$a_{n} = \frac{9}{10} - \frac{3}{20} (n - 1)$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Apply the distributive property.

$a_{n} = \frac{9}{10} - \frac{3}{20} n - \frac{3}{20} \cdot - 1$

Step 4.2

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

$a_{n} = \frac{9}{10} - \frac{n \cdot 3}{20} - \frac{3}{20} \cdot - 1$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Multiply $- 1$ by $- 1$ .

$a_{n} = \frac{9}{10} - \frac{n \cdot 3}{20} + 1 (\frac{3}{20})$

Step 4.3.2

Multiply $\frac{3}{20}$ by $1$ .

$a_{n} = \frac{9}{10} - \frac{n \cdot 3}{20} + \frac{3}{20}$

Step 4.4

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

$a_{n} = \frac{9}{10} - \frac{3 n}{20} + \frac{3}{20}$

Step 5

To write $\frac{9}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a_{n} = - \frac{3 n}{20} + \frac{9}{10} \cdot \frac{2}{2} + \frac{3}{20}$

Step 6

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

Tap for more steps...

Step 6.1

Multiply $\frac{9}{10}$ by $\frac{2}{2}$ .

$a_{n} = - \frac{3 n}{20} + \frac{9 \cdot 2}{10 \cdot 2} + \frac{3}{20}$

Step 6.2

Multiply $10$ by $2$ .

$a_{n} = - \frac{3 n}{20} + \frac{9 \cdot 2}{20} + \frac{3}{20}$

Step 7

Combine the numerators over the common denominator.

$a_{n} = - \frac{3 n}{20} + \frac{9 \cdot 2 + 3}{20}$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

Multiply $9$ by $2$ .

$a_{n} = - \frac{3 n}{20} + \frac{18 + 3}{20}$

Step 8.2

Add $18$ and $3$ .

$a_{n} = - \frac{3 n}{20} + \frac{21}{20}$

Step 9

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

$a_{5} = - \frac{3 (5)}{20} + \frac{21}{20}$

Step 10

Combine the numerators over the common denominator.

$a_{5} = \frac{- 3 \cdot 5 + 21}{20}$

Step 11

Simplify the expression.

Tap for more steps...

Step 11.1

Multiply $- 3$ by $5$ .

$a_{5} = \frac{- 15 + 21}{20}$

Step 11.2

Add $- 15$ and $21$ .

$a_{5} = \frac{6}{20}$

Step 12

Cancel the common factor of $6$ and $20$ .

Tap for more steps...

Step 12.1

Factor $2$ out of $6$ .

$a_{5} = \frac{2 (3)}{20}$

Step 12.2

Cancel the common factors.

Tap for more steps...

Step 12.2.1

Factor $2$ out of $20$ .

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

Step 12.2.2

Cancel the common factor.

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

Step 12.2.3

Rewrite the expression.

$a_{5} = \frac{3}{10}$