Find the Sum of the First 7 Terms 1/3 , 2/3 , 1 , 4/3 , 5/3

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

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.

Apply the distributive property.

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

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

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

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

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

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 $\frac{1}{3} + \frac{n}{3} - \frac{1}{3}$ .

Combine the numerators over the common denominator.

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

Subtract $1$ from $1$ .

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

Divide $0$ by $3$ .

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

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.

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

Step 8

Replace the variables with the known values to find $S_{7}$ .

$S_{7} = \frac{7}{2} \cdot (\frac{1}{3} + \frac{7}{3})$

Step 9

Combine the numerators over the common denominator.

$S_{7} = \frac{7}{2} \cdot \frac{1 + 7}{3}$

Step 10

Add $1$ and $7$ .

$S_{7} = \frac{7}{2} \cdot \frac{8}{3}$

Step 11

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$S_{7} = \frac{7}{2} \cdot \frac{2 (4)}{3}$

Cancel the common factor.

$S_{7} = \frac{7}{2} \cdot \frac{2 \cdot 4}{3}$

Rewrite the expression.

$S_{7} = 7 \cdot \frac{4}{3}$

Step 12

Combine $7$ and $\frac{4}{3}$ .

$S_{7} = \frac{7 \cdot 4}{3}$

Step 13

Multiply $7$ by $4$ .

$S_{7} = \frac{28}{3}$

Find the Sum of the First 7 Terms 1/3 , 2/3 , 1 , 4/3 , 5/3

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