Math Snap

STEP 1

1. The sequence given is a geometric series.
2. The general form of a geometric series is $a + ar + ar^2 + ar^3 + \ldots$.
3. The sum of an infinite geometric series $S$ is given by $S = \frac{a}{1-r}$, where $|r| < 1$.

STEP 2

1. Identify the first term $a$ and the common ratio $r$ of the series.
2. Determine the limit of the sequence term $a_n$ as $n \rightarrow \infty$.
3. Determine the sum of the infinite series $S_n$.

STEP 3

Identify the first term $a$ and the common ratio $r$ of the series:
The series is $6 + \frac{6}{5} + \frac{6}{25} + \frac{6}{125} + \ldots$.
The first term $a = 6$.
The common ratio $r$ is found by dividing the second term by the first term:
$$r = \frac{\frac{6}{5}}{6} = \frac{1}{5}$$

STEP 4

Determine the limit of the sequence term $a_n$ as $n \rightarrow \infty$:
The general term of the sequence $a_n = 6 \left(\frac{1}{5}\right)^{n-1}$.
As $n \rightarrow \infty$, the term $\left(\frac{1}{5}\right)^{n-1} \rightarrow 0$.
Thus, $\lim_{n \rightarrow \infty} a_n = 0$.

Determine the sum of the infinite series $S_n$:
The sum of an infinite geometric series is given by:
$$S = \frac{a}{1-r} = \frac{6}{1-\frac{1}{5}} = \frac{6}{\frac{4}{5}} = 6 \times \frac{5}{4} = \frac{30}{4} = 7.5$$ Thus, $\lim_{n \rightarrow \infty} S_n = 7.5$.
The completed blanks are:
1. $\lim_{n \rightarrow \infty} a_n = 0$
2. $\lim_{n \rightarrow \infty} S_n = 7.5$