Algebra Examples

$2$ , $1 \frac{3}{5}$ , $1 \frac{1}{5}$

Step 1

Convert $1 \frac{3}{5}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$2, 1 + \frac{3}{5}, 1 \frac{1}{5}$

Step 1.2

Add $1$ and $\frac{3}{5}$ .

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$2, \frac{5}{5} + \frac{3}{5}, 1 \frac{1}{5}$

Step 1.2.2

Combine the numerators over the common denominator.

$2, \frac{5 + 3}{5}, 1 \frac{1}{5}$

Step 1.2.3

Add $5$ and $3$ .

$2, \frac{8}{5}, 1 \frac{1}{5}$

Step 2

Convert $1 \frac{1}{5}$ to an improper fraction.

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Step 2.1

A mixed number is an addition of its whole and fractional parts.

$2, \frac{8}{5}, 1 + \frac{1}{5}$

Step 2.2

Add $1$ and $\frac{1}{5}$ .

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Step 2.2.1

Write $1$ as a fraction with a common denominator.

$2, \frac{8}{5}, \frac{5}{5} + \frac{1}{5}$

Step 2.2.2

Combine the numerators over the common denominator.

$2, \frac{8}{5}, \frac{5 + 1}{5}$

Step 2.2.3

Add $5$ and $1$ .

$2, \frac{8}{5}, \frac{6}{5}$

Step 3

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $- \frac{2}{5}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - \frac{2}{5}$

Step 4

This is the formula of an arithmetic sequence.

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

Step 5

Substitute in the values of $a_{1} = 2$ and $d = - \frac{2}{5}$ .

$a_{n} = 2 - \frac{2}{5} (n - 1)$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$a_{n} = 2 - \frac{2}{5} n - \frac{2}{5} \cdot - 1$

Step 6.2

Combine $n$ and $\frac{2}{5}$ .

$a_{n} = 2 - \frac{n \cdot 2}{5} - \frac{2}{5} \cdot - 1$

Step 6.3

Multiply $- \frac{2}{5} \cdot - 1$ .

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Step 6.3.1

Multiply $- 1$ by $- 1$ .

$a_{n} = 2 - \frac{n \cdot 2}{5} + 1 (\frac{2}{5})$

Step 6.3.2

Multiply $\frac{2}{5}$ by $1$ .

$a_{n} = 2 - \frac{n \cdot 2}{5} + \frac{2}{5}$

Step 6.4

Move $2$ to the left of $n$ .

$a_{n} = 2 - \frac{2 n}{5} + \frac{2}{5}$

Step 7

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$a_{n} = - \frac{2 n}{5} + 2 \cdot \frac{5}{5} + \frac{2}{5}$

Step 8

Combine $2$ and $\frac{5}{5}$ .

$a_{n} = - \frac{2 n}{5} + \frac{2 \cdot 5}{5} + \frac{2}{5}$

Step 9

Combine the numerators over the common denominator.

$a_{n} = - \frac{2 n}{5} + \frac{2 \cdot 5 + 2}{5}$

Step 10

Simplify the numerator.

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Step 10.1

Multiply $2$ by $5$ .

$a_{n} = - \frac{2 n}{5} + \frac{10 + 2}{5}$

Step 10.2

Add $10$ and $2$ .

$a_{n} = - \frac{2 n}{5} + \frac{12}{5}$