To represent the sum [tex]\(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17\)[/tex] using summation notation, we need to identify a general term for this sequence and the appropriate limits of summation.
First, let's observe the given sequence: [tex]\(1, 3, 5, 7, 9, 11, 13, 15, 17\)[/tex].
It is clear that this is an arithmetic sequence with:
- The first term [tex]\(a = 1\)[/tex]
- Common difference [tex]\(d = 2\)[/tex]
The general term of an arithmetic sequence can be expressed as:
[tex]\[ a_n = a + (n-1)d \][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex], we get:
[tex]\[ a_n = 1 + (n-1) \cdot 2 \][/tex]
[tex]\[ a_n = 1 + 2n - 2 \][/tex]
[tex]\[ a_n = 2n - 1 \][/tex]
Now, we need to determine the limits for [tex]\(n\)[/tex]. Observing the sequence, the 1st term [tex]\(a_1\)[/tex] corresponds to [tex]\(1\)[/tex], the 2nd term [tex]\(a_2\)[/tex] corresponds to [tex]\(3\)[/tex], and so forth. The 9th term [tex]\(a_9\)[/tex] corresponds to [tex]\(17\)[/tex]. Hence, [tex]\(n\)[/tex] ranges from [tex]\(1\)[/tex] to [tex]\(9\)[/tex].
Therefore, the summation notation can be written as:
[tex]\[ \sum_{n=1}^{9} (2n - 1) \][/tex]
Thus, the summation notation that best represents the sum [tex]\(1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17\)[/tex] is:
[tex]\[ \sum_{n=1}^{9} (2n - 1) \][/tex]
