Algebra Examples

$\frac{7}{2}$ , $\frac{9}{4}$ , $1$ , $- \frac{1}{4}$ , $- \frac{3}{2}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $- \frac{5}{4}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - \frac{5}{4}$

Step 2

This is the formula of an arithmetic sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{7}{2}$ and $d = - \frac{5}{4}$ .

$a_{n} = \frac{7}{2} - \frac{5}{4} (n - 1)$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$a_{n} = \frac{7}{2} - \frac{5}{4} n - \frac{5}{4} \cdot - 1$

Step 4.2

Combine $n$ and $\frac{5}{4}$ .

$a_{n} = \frac{7}{2} - \frac{n \cdot 5}{4} - \frac{5}{4} \cdot - 1$

Step 4.3

Multiply $- \frac{5}{4} \cdot - 1$ .

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$a_{n} = \frac{7}{2} - \frac{n \cdot 5}{4} + 1 (\frac{5}{4})$

Step 4.3.2

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

$a_{n} = \frac{7}{2} - \frac{n \cdot 5}{4} + \frac{5}{4}$

Step 4.4

Move $5$ to the left of $n$ .

$a_{n} = \frac{7}{2} - \frac{5 n}{4} + \frac{5}{4}$

Step 5

To write $\frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a_{n} = - \frac{5 n}{4} + \frac{7}{2} \cdot \frac{2}{2} + \frac{5}{4}$

Step 6

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{7}{2}$ by $\frac{2}{2}$ .

$a_{n} = - \frac{5 n}{4} + \frac{7 \cdot 2}{2 \cdot 2} + \frac{5}{4}$

Step 6.2

Multiply $2$ by $2$ .

$a_{n} = - \frac{5 n}{4} + \frac{7 \cdot 2}{4} + \frac{5}{4}$

Step 7

Combine the numerators over the common denominator.

$a_{n} = - \frac{5 n}{4} + \frac{7 \cdot 2 + 5}{4}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $7$ by $2$ .

$a_{n} = - \frac{5 n}{4} + \frac{14 + 5}{4}$

Step 8.2

Add $14$ and $5$ .

$a_{n} = - \frac{5 n}{4} + \frac{19}{4}$

Step 9

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

$a_{6} = - \frac{5 (6)}{4} + \frac{19}{4}$

Step 10

Combine the numerators over the common denominator.

$a_{6} = \frac{- 5 \cdot 6 + 19}{4}$

Step 11

Simplify the expression.

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Step 11.1

Multiply $- 5$ by $6$ .

$a_{6} = \frac{- 30 + 19}{4}$

Step 11.2

Add $- 30$ and $19$ .

$a_{6} = \frac{- 11}{4}$

Step 11.3

Move the negative in front of the fraction.

$a_{6} = - \frac{11}{4}$