Problem:

Given the geometric sequence:

$5, 1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \dots$

We are tasked to:

1. Find the 14th term of the sequence.
2. Find the sum of the sequence (if possible).

Solution:

Step 1: General formula for a geometric sequence

In a geometric sequence, the $n$-th term ($a_n$) is given by the formula:

$a_n = a_1 \cdot r^{n-1}$

where:

• $a_1$ is the first term,
• $r$ is the common ratio between consecutive terms,
• $n$ is the term number.

Step 2: Identifying the values

• The first term $a_1 = 5$.
• The common ratio $r$ is the ratio between any two consecutive terms. For this sequence, the ratio between the second term and the first term is:

$r = \frac{1}{5} \div 5 = \frac{1}{5}$

Thus, $r = \frac{1}{5}$.

Step 3: Finding the 14th term

To find the 14th term ($a_{14}$), we use the formula:

$a_{14} = 5 \cdot \left( \frac{1}{5} \right)^{14-1}$ $a_{14} = 5 \cdot \left( \frac{1}{5} \right)^{13}$ $a_{14} = 5 \cdot \frac{1}{5^{13}}$ $a_{14} = \frac{5}{5^{13}} = \frac{1}{5^{12}}$

So, the 14th term is $\frac{1}{5^{12}}$.

Step 4: Finding the sum of the sequence

The sum $S_n$ of the first $n$ terms of a geometric series is given by the formula:

$S_n = \frac{a_1 (1 - r^n)}{1 - r}$

for $|r| < 1$. In this case, $r = \frac{1}{5}$, which satisfies $|r| < 1$.

Now, to find the sum of the first 14 terms:

$S_{14} = \frac{5 \left( 1 - \left( \frac{1}{5} \right)^{14} \right)}{1 - \frac{1}{5}}$ $S_{14} = \frac{5 \left( 1 - \frac{1}{5^{14}} \right)}{\frac{4}{5}}$ $S_{14} = \frac{5}{\frac{4}{5}} \left( 1 - \frac{1}{5^{14}} \right)$ $S_{14} = \frac{25}{4} \left( 1 - \frac{1}{5^{14}} \right)$

Since $\frac{1}{5^{14}}$ is very small, we can approximate:

$S_{14} \approx \frac{25}{4} \cdot 1 = \frac{25}{4}$

Thus, the sum of the first 14 terms is approximately $\frac{25}{4}$.

Final Answers:

• The 14th term is $\frac{1}{5^{12}}$.
• The sum of the first 14 terms is approximately $\frac{25}{4}$.

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Would you like further clarification on the steps, or do you have any questions?

Here are 5 related questions for you:

1. How do you find the sum of an infinite geometric series?
2. What happens to the sum of a geometric series if $r$ is greater than 1?
3. How do you find the common ratio in a geometric sequence?
4. Can a geometric series have a negative common ratio? If so, how does this affect the sequence?
5. What is the effect of a negative common ratio on the terms of the geometric sequence?

Tip: To better understand geometric sequences, try plotting the terms of the sequence to visualize how the terms decrease rapidly when $r$ is a fraction.