Math Snap

STEP 1

What is this asking?
We need to find the sum of a geometric series and express the result in a specific form.
Watch out!
It's easy to mess up the formula for the sum of a geometric series if we're not careful!
Also, remember that the first term isn't always the same as the common ratio.

STEP 2

1. Identify the series
2. Apply the formula
3. Simplify the result

STEP 3

Alright, let's dive in!
We've got a geometric series here.
How do we know?
Each term is obtained by multiplying the previous term by the same number.
That number is our common ratio, often denoted by $r$.
In our case, $r = 4$.
See how each term is 4 times the previous one?

STEP 4

Our first term, often denoted by $a$, is $\frac{1}{5}$.
So, we have $a = \frac{1}{5}$.

STEP 5

We also know that there are $n$ terms in this series.

STEP 6

The sum of a finite geometric series is given by the formula:
$$S_n = \frac{a(r^n - 1)}{r - 1}$$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

STEP 7

Let's plug in our values: $a = \frac{1}{5}$, $r = 4$, and we have $n$ terms.
$$S_n = \frac{\frac{1}{5}(4^n - 1)}{4 - 1}$$

STEP 8

Let's simplify the denominator: $4 - 1 = 3$.
$$S_n = \frac{\frac{1}{5}(4^n - 1)}{3}$$

STEP 9

Now, let's multiply the fraction by $\frac{1}{3}$, which is the same as dividing by 3.
Remember, dividing by 3 is the same as multiplying by its reciprocal, $\frac{1}{3}$.
$$S_n = \frac{1}{5} \cdot \frac{1}{3} (4^n - 1)$$

STEP 10

Multiplying the fractions $\frac{1}{5}$ and $\frac{1}{3}$ gives us:
$$S_n = \frac{1}{15}(4^n - 1)$$

The sum is $S_n = \frac{1}{15}(4^n - 1)$.