Algebra Examples

$- 1$ , $\frac{2}{3}$ , $\frac{7}{3}$ , $4$ , $\frac{17}{3}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $\frac{5}{3}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = \frac{5}{3}$

Step 2

This is the formula of an arithmetic sequence.

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

Step 3

Substitute in the values of $a_{1} = - 1$ and $d = \frac{5}{3}$ .

$a_{n} = - 1 + \frac{5}{3} (n - 1)$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine $\frac{5}{3}$ and $n$ .

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

Step 4.3

Multiply $\frac{5}{3} \cdot - 1$ .

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Step 4.3.1

Combine $\frac{5}{3}$ and $- 1$ .

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

Step 4.3.2

Multiply $5$ by $- 1$ .

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $3$ .

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

Step 8.2

Subtract $5$ from $- 3$ .

$a_{n} = \frac{5 n}{3} + \frac{- 8}{3}$

Step 9

Move the negative in front of the fraction.

$a_{n} = \frac{5 n}{3} - \frac{8}{3}$

Step 10

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

$a_{7} = \frac{5 (7)}{3} - \frac{8}{3}$

Step 11

Combine the numerators over the common denominator.

$a_{7} = \frac{5 (7) - 8}{3}$

Step 12

Simplify the expression.

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Step 12.1

Multiply $5$ by $7$ .

$a_{7} = \frac{35 - 8}{3}$

Step 12.2

Subtract $8$ from $35$ .

$a_{7} = \frac{27}{3}$

Step 12.3

Divide $27$ by $3$ .

$a_{7} = 9$