Find the Sum of the Series 3+1+1/3+1/9+1/27

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by ${\displaystyle \frac{1}{3}}$ gives the next term. In other words, ${\displaystyle a_n=a_1{r}^{n-1}}$.

This is the form of a geometric sequence.

Substitute in the values of ${\displaystyle a_1=3}$ and ${\displaystyle r=\frac{1}{3}}$.

Apply the product rule to ${\displaystyle \frac{1}{3}}$.

One to any power is one.

Combine ${\displaystyle 3}$ and ${\displaystyle \frac{1}{3^{n-1}}}$.

This is the formula to find the sum of the first ${\displaystyle n}$ terms of the geometric sequence. To evaluate it, find the values of ${\displaystyle r}$ and ${\displaystyle a_1}$.

Replace the variables with the known values to find ${\displaystyle S_5}$.

Dividing two negative values results in a positive value.

Multiply the numerator by the reciprocal of the denominator.