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]
