Algebra Examples

$\frac{1}{3}$ , $\frac{2}{3}$ , $1$ , $\frac{4}{3}$ , $\frac{5}{3}$

Step 1

This is the formula to find the sum of the first $n$ terms of the sequence. To evaluate it, the values of the first and $n$ th terms must be found.

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

Step 2

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

Arithmetic Sequence: $d = \frac{1}{3}$

Step 3

This is the formula of an arithmetic sequence.

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

Step 4

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

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

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Move the negative in front of the fraction.

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

Step 6

Combine the opposite terms in $\frac{1}{3} + \frac{n}{3} - \frac{1}{3}$ .

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Step 6.1

Combine the numerators over the common denominator.

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

Step 6.2

Subtract $1$ from $1$ .

$a_{n} = \frac{n}{3} + \frac{0}{3}$

Step 6.3

Divide $0$ by $3$ .

$a_{n} = \frac{n}{3} + 0$

Step 6.4

Add $\frac{n}{3}$ and $0$ .

$a_{n} = \frac{n}{3}$

Step 7

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

$a_{6} = \frac{6}{3}$

Step 8

Divide $6$ by $3$ .

$a_{6} = 2$

Step 9

Replace the variables with the known values to find $S_{6}$ .

$S_{6} = \frac{6}{2} \cdot (\frac{1}{3} + 2)$

Step 10

Divide $6$ by $2$ .

$S_{6} = 3 \cdot (\frac{1}{3} + 2)$

Step 11

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$S_{6} = 3 \cdot (\frac{1}{3} + 2 \cdot \frac{3}{3})$

Step 12

Combine $2$ and $\frac{3}{3}$ .

$S_{6} = 3 \cdot (\frac{1}{3} + \frac{2 \cdot 3}{3})$

Step 13

Combine the numerators over the common denominator.

$S_{6} = 3 \cdot \frac{1 + 2 \cdot 3}{3}$

Step 14

Simplify the numerator.

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Step 14.1

Multiply $2$ by $3$ .

$S_{6} = 3 \cdot \frac{1 + 6}{3}$

Step 14.2

Add $1$ and $6$ .

$S_{6} = 3 \cdot \frac{7}{3}$

Step 15

Cancel the common factor of $3$ .

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Step 15.1

Cancel the common factor.

$S_{6} = 3 \cdot \frac{7}{3}$

Step 15.2

Rewrite the expression.

$S_{6} = 7$