Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To determine which rule defines the sum [tex]\( S_n \)[/tex] of the geometric series for [tex]\( a_n = 3 \cdot (0.7)^n \)[/tex], we can use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series. The sum of the first [tex]\( n \)[/tex] terms [tex]\( S_n \)[/tex] of a geometric series with the first term [tex]\( a \)[/tex] and common ratio [tex]\( r \)[/tex] is given by:
[tex]\[ S_n = a \frac{1-r^{n+1}}{1-r} \][/tex]
In our case:
- The first term [tex]\( a = 3 \cdot (0.7)^0 = 3 \)[/tex]
- The common ratio [tex]\( r = 0.7 \)[/tex]
Using the formula, we substitute [tex]\( a \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ S_n = 3 \cdot \frac{1 - (0.7)^{n+1}}{1 - 0.7} \][/tex]
Simplify the expression:
[tex]\[ S_n = 3 \cdot \frac{1 - (0.7)^{n+1}}{0.3} \][/tex]
[tex]\[ S_n = 3 \cdot \frac{1}{0.3} - 3 \cdot \frac{(0.7)^{n+1}}{0.3} \][/tex]
[tex]\[ S_n = 10 - 10 \cdot (0.7)^{n+1} \][/tex]
Now, we can determine which one of the given choices matches this result:
1. [tex]\( S_n = 7 \left(0.7^n\right) \)[/tex]
2. [tex]\( S_n = 2.1 \left(0.7^n\right) \)[/tex]
3. [tex]\( S_n = 7 \left(1 - 0.7^n\right) \)[/tex]
4. [tex]\( S_n = 2.1 \left(1 - 0.7^n\right) \)[/tex]
Notice that neither of these matches directly with our derived expression [tex]\[ 10 - 10 \cdot (0.7)^{n+1} \][/tex]. However, we also consider simpler steps that can point to an infinite series sum.
For an infinite geometric series, the sum is given by:
[tex]\[ S = \frac{a}{1-r} \][/tex]
In our case, substituting [tex]\( a = 3 \)[/tex] and [tex]\( r = 0.7 \)[/tex]:
[tex]\[ S = \frac{3}{1-0.7} = \frac{3}{0.3} = 10 \][/tex]
By checking through possibilities given for the defined clues in question:
Let’s evaluate given options recognizing their difference to geometric infinite summation directly.
1. [tex]\( S_n = 7 \left(0.7^n\right) \)[/tex]: This doesn’t fit geometric sum pattern.
2. [tex]\( S_n = 2.1 \left(0.7^n\right) \)[/tex]: This clearly doesn’t fit geometric sum pattern.
3. [tex]\( S_n = 7 \left(1 - 0.7^n\right) \)[/tex]: This matches partial sum form.
4. [tex]\( S_n = 2.1 \left(1 - 0.7^n\right) \)[/tex]: also doesn’t match comprehensive sum.
Only expression involving form similar to geometric finite summation (similar to [tex]\( 1-r^{n+1}\)[/tex]/formation proposed rightly evaluated fits derivable terms i.e in option 3.
Therefore, the rule that defines [tex]\( S_n \)[/tex] is:
[tex]\[ S_n = 7 \left(1-0.7^n\right) \][/tex]
[tex]\[ S_n = a \frac{1-r^{n+1}}{1-r} \][/tex]
In our case:
- The first term [tex]\( a = 3 \cdot (0.7)^0 = 3 \)[/tex]
- The common ratio [tex]\( r = 0.7 \)[/tex]
Using the formula, we substitute [tex]\( a \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ S_n = 3 \cdot \frac{1 - (0.7)^{n+1}}{1 - 0.7} \][/tex]
Simplify the expression:
[tex]\[ S_n = 3 \cdot \frac{1 - (0.7)^{n+1}}{0.3} \][/tex]
[tex]\[ S_n = 3 \cdot \frac{1}{0.3} - 3 \cdot \frac{(0.7)^{n+1}}{0.3} \][/tex]
[tex]\[ S_n = 10 - 10 \cdot (0.7)^{n+1} \][/tex]
Now, we can determine which one of the given choices matches this result:
1. [tex]\( S_n = 7 \left(0.7^n\right) \)[/tex]
2. [tex]\( S_n = 2.1 \left(0.7^n\right) \)[/tex]
3. [tex]\( S_n = 7 \left(1 - 0.7^n\right) \)[/tex]
4. [tex]\( S_n = 2.1 \left(1 - 0.7^n\right) \)[/tex]
Notice that neither of these matches directly with our derived expression [tex]\[ 10 - 10 \cdot (0.7)^{n+1} \][/tex]. However, we also consider simpler steps that can point to an infinite series sum.
For an infinite geometric series, the sum is given by:
[tex]\[ S = \frac{a}{1-r} \][/tex]
In our case, substituting [tex]\( a = 3 \)[/tex] and [tex]\( r = 0.7 \)[/tex]:
[tex]\[ S = \frac{3}{1-0.7} = \frac{3}{0.3} = 10 \][/tex]
By checking through possibilities given for the defined clues in question:
Let’s evaluate given options recognizing their difference to geometric infinite summation directly.
1. [tex]\( S_n = 7 \left(0.7^n\right) \)[/tex]: This doesn’t fit geometric sum pattern.
2. [tex]\( S_n = 2.1 \left(0.7^n\right) \)[/tex]: This clearly doesn’t fit geometric sum pattern.
3. [tex]\( S_n = 7 \left(1 - 0.7^n\right) \)[/tex]: This matches partial sum form.
4. [tex]\( S_n = 2.1 \left(1 - 0.7^n\right) \)[/tex]: also doesn’t match comprehensive sum.
Only expression involving form similar to geometric finite summation (similar to [tex]\( 1-r^{n+1}\)[/tex]/formation proposed rightly evaluated fits derivable terms i.e in option 3.
Therefore, the rule that defines [tex]\( S_n \)[/tex] is:
[tex]\[ S_n = 7 \left(1-0.7^n\right) \][/tex]
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
