Certainly! Let's go through each series in detail and represent them using summation notation.
### Series 9: [tex]\(5 + 6 + 7 + \cdots + 12\)[/tex]
This is an arithmetic series where the first term ([tex]\(a\)[/tex]) is 5 and the last term is 12. The common difference ([tex]\(d\)[/tex]) is 1.
The series can be written in summation notation as:
[tex]\[
\sum_{k=0}^{7} (5 + k)
\][/tex]
or
[tex]\[
\sum_{k=5}^{12} k
\][/tex]
### Series 10: [tex]\(-3, -6, -9, \ldots\)[/tex]
This is another arithmetic series, but with a common difference of -3. The first term ([tex]\(a\)[/tex]) is -3.
The [tex]\(k\)[/tex]-th term ([tex]\(a_k\)[/tex]) is given by:
[tex]\[
a_k = -3k
\][/tex]
The series can be written in summation notation as:
[tex]\[
\sum_{k=1}^{n} (-3k)
\][/tex]
### Series 11: [tex]\(\frac{1}{3} + \frac{2}{4} + \frac{3}{5} + \cdots + \frac{12}{14}\)[/tex]
This series is somewhat more intricate, involving fractions where the numerator is [tex]\(k\)[/tex] and the denominator is [tex]\(k + 2\)[/tex] for [tex]\(k\)[/tex] ranging from 1 to 12.
The series can be written in summation notation as:
[tex]\[
\sum_{k=1}^{12} \frac{k}{k+2}
\][/tex]
### Series 12: [tex]\(3 + 5 + 7 + \ldots\)[/tex]
This is another arithmetic series with a common difference of 2. The first term ([tex]\(a\)[/tex]) is 3.
The [tex]\(k\)[/tex]-th term ([tex]\(a_k\)[/tex]) is given by:
[tex]\[
a_k = 3 + (k-1) \cdot 2
\][/tex]
The series can be written in summation notation as:
[tex]\[
\sum_{k=1}^{n} (3 + 2(k-1))
\][/tex]
For specific cases, for example, up to [tex]\(n\)[/tex] terms, you would sum up from [tex]\(k=1\)[/tex] to that specific [tex]\(n\)[/tex].
These representations provide a clear mathematical notation for each series and make them easier to analyze and compute.
