To determine the value of [tex]\( r \)[/tex] for the given geometric series, we need to understand the structure of the sum.
The general form of a geometric series is:
[tex]\[ \sum_{n=1}^k a \cdot r^{n-1} \][/tex]
Where:
- [tex]\( a \)[/tex] is the first term of the series.
- [tex]\( r \)[/tex] is the common ratio.
- [tex]\( n \)[/tex] is the term number.
- [tex]\( k \)[/tex] is the number of terms.
Given the series:
[tex]\[ \sum_{n=1}^3 1.3(0.8)^{n-1} \][/tex]
To identify the values:
- [tex]\( a \)[/tex] is the coefficient just before the ratio term, which is [tex]\( 1.3 \)[/tex].
- The expression being raised to the power [tex]\( (n-1) \)[/tex] is [tex]\( 0.8 \)[/tex].
From this structure of the series, [tex]\( r \)[/tex] is the common ratio, which is the value that each term is multiplied by to get the next term.
Since the expression [tex]\( (0.8)^{n-1} \)[/tex] takes the form of [tex]\( r^{n-1} \)[/tex] in the general formula for a geometric series, we can see that [tex]\( r = 0.8 \)[/tex].
Therefore, the value of [tex]\( r \)[/tex] for this geometric series is:
[tex]\[ \boxed{0.8} \][/tex]
