Find the sum of first $n$ terms and the sum of first 5 terms of the geometric series $1+\frac{2}{3}+\frac{4}{9}+\ldots$

Step 1: Identify the first term and common ratio

The given series is a geometric series. The first term, denoted by $a$, is the first number in the series, which is 1. The common ratio, denoted by $r$, is found by dividing any term by its preceding term. Here, we divide the second term by the first term: $\frac{2/3}{1} = \frac{2}{3}$.

Step 2: Formula for the sum of first n terms

The formula for the sum of the first $n$ terms of a geometric series is $S_n = \frac{a(1-r^n)}{1-r}$, provided that the absolute value of the common ratio $r$ is less than 1. In this case, $|2/3| < 1$, so this formula is applicable.

Step 3: Calculate the sum of first n terms

Substitute the values of $a=1$ and $r=\frac{2}{3}$ into the formula for $S_n$. Simplify the expression by first calculating the denominator $1 - \frac{2}{3} = \frac{1}{3}$, and then multiplying the numerator by the reciprocal of the denominator.

Step 4: Calculate the sum of first 5 terms

To find the sum of the first 5 terms, substitute $n=5$ into the derived formula for $S_n$. Calculate $(\frac{2}{3})^5 = \frac{2^5}{3^5} = \frac{32}{243}$. Then, perform the subtraction inside the parenthesis and multiply by 3 to get the final sum.

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