Certainly! Let's go through the step-by-step process to find the terms and the sum of the given geometric series, [tex]\(5000 + 4000 + 3200 + \ldots\)[/tex], for 5 terms.
### Step 1: Identify the first term and common ratio
First, identify the first term ([tex]\(a_1\)[/tex]) and the common ratio ([tex]\(r\)[/tex]).
- The first term [tex]\(a_1 = 5000\)[/tex].
- To find the common ratio, we divide the second term by the first term:
[tex]\[
r = \frac{4000}{5000} = 0.8
\][/tex]
### Step 2: Calculate each term in the series
Using the first term and the common ratio, we can calculate each subsequent term in the series.
#### Second term ([tex]\(a_2\)[/tex]):
[tex]\[
a_2 = a_1 \cdot r = 5000 \cdot 0.8 = 4000
\][/tex]
#### Third term ([tex]\(a_3\)[/tex]):
[tex]\[
a_3 = a_2 \cdot r = 4000 \cdot 0.8 = 3200
\][/tex]
#### Fourth term ([tex]\(a_4\)[/tex]):
[tex]\[
a_4 = a_3 \cdot r = 3200 \cdot 0.8 = 2560
\][/tex]
#### Fifth term ([tex]\(a_5\)[/tex]):
[tex]\[
a_5 = a_4 \cdot r = 2560 \cdot 0.8 = 2048
\][/tex]
So, the five terms of the series are:
[tex]\[
5000, 4000, 3200, 2560, 2048
\][/tex]
### Step 3: Calculate the sum of the series
The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric series can be calculated using the formula:
[tex]\[
S_n = a_1 \frac{1 - r^n}{1 - r}
\][/tex]
For [tex]\(n = 5\)[/tex], the terms we need are:
[tex]\[
a_1 = 5000, \quad r = 0.8, \quad n = 5
\][/tex]
Plugging these values into the sum formula:
[tex]\[
S_5 = 5000 \frac{1 - (0.8)^5}{1 - 0.8}
\][/tex]
### Calculations:
[tex]\[
(0.8)^5 = 0.32768
\][/tex]
[tex]\[
1 - 0.32768 = 0.67232
\][/tex]
[tex]\[
1 - 0.8 = 0.2
\][/tex]
[tex]\[
S_5 = 5000 \frac{0.67232}{0.2} = 5000 \cdot 3.3616 = 16808
\][/tex]
### Conclusion
The first five terms of the series are:
[tex]\[
5000, 4000, 3200, 2560, 2048
\][/tex]
The sum of these five terms is:
[tex]\[
16808
\][/tex]
Therefore, the complete answer is:
[tex]\[
\boxed{5000, 4000, 3200, 2560, 2048}
\][/tex]
[tex]\[
\boxed{16808}
\][/tex]
