Let's consider the series [tex]\(-2, 6, -18, 54, \ldots\)[/tex]. To find the sum of the first 20 terms, we'll follow these steps:
1. Identify the type of series: This is a geometric series because each term is a constant multiple (common ratio [tex]\( r \)[/tex]) of the previous one.
2. Determine the first term ([tex]\( a \)[/tex]) and the common ratio ([tex]\( r \)[/tex]):
- The first term ([tex]\( a \)[/tex]) of the series is [tex]\(-2\)[/tex].
- The common ratio ([tex]\( r \)[/tex]) can be found by dividing the second term by the first term:
[tex]\[
r = \frac{6}{-2} = -3
\][/tex]
3. Set up the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series:
The sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric series can be found using the formula:
[tex]\[
S_n = a \frac{1 - r^n}{1 - r}
\][/tex]
where:
- [tex]\( a \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio, and
- [tex]\( n \)[/tex] is the number of terms.
4. Plug in the values we have:
Here, [tex]\( a = -2 \)[/tex], [tex]\( r = -3 \)[/tex], and [tex]\( n = 20 \)[/tex]. Plugging these into the formula, we get:
[tex]\[
S_{20} = -2 \frac{1 - (-3)^{20}}{1 - (-3)}
\][/tex]
5. Simplify the expression:
Let's simplify within the fraction:
[tex]\[
S_{20} = -2 \frac{1 - (-3)^{20}}{1 + 3}
\][/tex]
[tex]\[
S_{20} = -2 \frac{1 - (-3)^{20}}{4}
\][/tex]
6. Final computation:
- Calculate [tex]\( (-3)^{20} \)[/tex] (a large number, but crucial for the formula),
- Subtract that result from 1,
- Divide that by 4,
- Multiply by [tex]\(-2\)[/tex] to get the final sum.
7. Result:
After performing these calculations, we find that the sum of the first 20 terms of the series is approximately:
[tex]\[
1,743,392,200
\][/tex]
So, the sum of the first 20 terms of the series [tex]\(-2, 6, -18, 54, \ldots\)[/tex] is [tex]\(\boxed{1,743,392,200}\)[/tex].
