To determine the values of [tex]\( r \)[/tex] and [tex]\( a_1 \)[/tex] for the geometric series [tex]\(\sum_{k=1}^6 \frac{1}{4}(2)^{k-1}\)[/tex], let's break down the series into its components.
The general form of a geometric series is given by:
[tex]\[ \sum_{k=1}^n a \cdot r^{k-1} \][/tex]
where:
- [tex]\( a \)[/tex] is the first term of the series ([tex]\(a_1\)[/tex]).
- [tex]\( r \)[/tex] is the common ratio, which is the factor by which each term is multiplied to get the next term.
For the series [tex]\(\sum_{k=1}^6 \frac{1}{4}(2)^{k-1}\)[/tex]:
- The first term [tex]\(a_1\)[/tex] is the coefficient of the series when [tex]\( k = 1 \)[/tex], which is [tex]\(\frac{1}{4}\)[/tex]. This means [tex]\(a_1 = \frac{1}{4}\)[/tex].
- The common ratio [tex]\(r\)[/tex] is the factor that each term is multiplied by to get the next term. Here, every term is multiplied by [tex]\(2\)[/tex], so [tex]\(r = 2\)[/tex].
Hence, the correct values are:
[tex]\[ r = 2 \][/tex]
[tex]\[ a_1 = \frac{1}{4} \][/tex]
Therefore, the correct choice from the given options is:
[tex]\[ r=2; \, a_1=\frac{1}{4} \][/tex]
