Find the 6th Term 5/6 , 2/3 , 1/2 , 1/3

$\frac{5}{6}$ , $\frac{2}{3}$ , $\frac{1}{2}$ , $\frac{1}{3}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $- \frac{1}{6}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - \frac{1}{6}$

Step 2

This is the formula of an arithmetic sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{5}{6}$ and $d = - \frac{1}{6}$ .

$a_{n} = \frac{5}{6} - \frac{1}{6} (n - 1)$

Step 4

Simplify each term.

Apply the distributive property.

$a_{n} = \frac{5}{6} - \frac{1}{6} n - \frac{1}{6} \cdot - 1$

Combine $n$ and $\frac{1}{6}$ .

$a_{n} = \frac{5}{6} - \frac{n}{6} - \frac{1}{6} \cdot - 1$

Multiply $- \frac{1}{6} \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$a_{n} = \frac{5}{6} - \frac{n}{6} + 1 (\frac{1}{6})$

Multiply $\frac{1}{6}$ by $1$ .

$a_{n} = \frac{5}{6} - \frac{n}{6} + \frac{1}{6}$

Step 5

Combine the numerators over the common denominator.

$a_{n} = \frac{5 - n + 1}{6}$

Step 6

Add $5$ and $1$ .

$a_{n} = \frac{- n + 6}{6}$

Step 7

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

$a_{6} = \frac{- (6) + 6}{6}$

Step 8

Cancel the common factor of $- (6) + 6$ and $6$ .

Factor $6$ out of $- (6)$ .

$a_{6} = \frac{6 \cdot - 1 + 6}{6}$

Factor $6$ out of $6$ .

$a_{6} = \frac{6 \cdot - 1 + 6 \cdot 1}{6}$

Factor $6$ out of $6 \cdot - 1 + 6 (1)$ .

$a_{6} = \frac{6 \cdot (- 1 + 1)}{6}$

Cancel the common factors.

Factor $6$ out of $6$ .

$a_{6} = \frac{6 \cdot (- 1 + 1)}{6 (1)}$

Cancel the common factor.

$a_{6} = \frac{6 \cdot (- 1 + 1)}{6 \cdot 1}$

Rewrite the expression.

$a_{6} = \frac{- 1 + 1}{1}$

Divide $- 1 + 1$ by $1$ .

$a_{6} = - 1 + 1$

Step 9

Add $- 1$ and $1$ .

$a_{6} = 0$

Find the 6th Term 5/6 , 2/3 , 1/2 , 1/3

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