To find how many dots we would draw in the first 15 stages, let's analyze each of the provided expressions:
1. Expression: [tex]\( 3\left(1+3+3^2+\ldots+3^{14}\right) \)[/tex]
This expression represents [tex]\( 3 \)[/tex] times the sum of a geometric series from [tex]\( 3^0 \)[/tex] to [tex]\( 3^{14} \)[/tex]. The sum of a geometric series where the first term [tex]\( a = 1 \)[/tex] and the common ratio [tex]\( r = 3 \)[/tex] is given by:
[tex]\[
S = \frac{r^{n} - 1}{r - 1}
\][/tex]
Here, [tex]\( n = 15 \)[/tex] (since we have 15 terms from [tex]\( 3^0 \)[/tex] to [tex]\( 3^{14} \)[/tex]). Thus,
[tex]\[
S = \frac{3^{15} - 1}{3 - 1}
\][/tex]
Therefore, [tex]\( 3 \left(1 + 3 + 3^2 + \ldots + 3^{14}\right) = 3 \times \frac{3^{15} - 1}{3 - 1} \)[/tex].
The numerical value equates to [tex]\( 21,523,359 \)[/tex].
2. Expression: [tex]\(\frac{1-3^{15}}{1-3}\)[/tex]
Using the formula for the sum of a geometric series:
[tex]\[
\frac{1 - 3^{15}}{1 - 3}
\][/tex]
This directly computes the sum of the series from [tex]\( 3^0 \)[/tex] to [tex]\( 3^{14} \)[/tex]. The numerical value is [tex]\( 7,174,453 \)[/tex].
3. Expression: [tex]\((1-3)\left(1+3+3^2+\ldots+3^{14}\right)\)[/tex]
The common ratio [tex]\( (1 - 3) \)[/tex] is [tex]\(-2\)[/tex]. Thus, this expression represents:
[tex]\[
-2 \times \sum_{i=0}^{14} 3^i
\][/tex]
It is incorrect because multiplying by [tex]\(-2\)[/tex] does not represent the number of dots correctly.
4. Expression: [tex]\(3 \frac{1-3^{15}}{1-3}\)[/tex]
Based on the first expression's explanation and knowing:
[tex]\[
\frac{1 - 3^{15}}{1 - 3} = \sum_{i=0}^{14} 3^i
\][/tex]
This expression represents:
[tex]\[
3 \times \left( \frac{1 - 3^{15}}{1 - 3} \right)
\][/tex]
This is exactly the same calculation as the first expression. The numerical value is [tex]\( 21,523,359 \)[/tex].
5. Expression: [tex]\(3(1-3)\left(1+3+3^2+\ldots+3^{14}\right)\)[/tex]
Following similar logic as the third expression, [tex]\( (1 - 3) = -2 \)[/tex]:
[tex]\[
3(-2) \left( \sum_{i=0}^{14} 3^i \right) = -6 \sum_{i=0}^{14} 3^i
\][/tex]
This is also incorrect, as it wrongly moves the ratio and initial terms.
Now, let's list the correct expressions representing the number of dots:
1. [tex]\( 3\left(1 + 3 + 3^2 + \ldots + 3^{14}\right) \)[/tex]
2. [tex]\( 3 \frac{1 - 3^{15}}{1 - 3} \)[/tex]
Both expressions correctly tally up to [tex]\( 21,523,359 \)[/tex]. The numerical value confirms this conclusion based on appropriate series manipulations.
