Sure, let's find the sum of the arithmetic sequence [tex]\(9, 10, 11, \ldots, n+8\)[/tex] step by step.
### Step 1: Identify the parameters of the arithmetic sequence
An arithmetic sequence has the form:
[tex]\[ a, a+d, a+2d, \ldots, \][/tex]
Where:
- [tex]\(a\)[/tex] is the first term
- [tex]\(d\)[/tex] is the common difference between the terms
- [tex]\(l\)[/tex] is the last term
In this problem:
- First term [tex]\(a = 9\)[/tex]
- Common difference [tex]\(d = 1\)[/tex]
- Last term [tex]\(l = n + 8\)[/tex]
### Step 2: Find the number of terms in the sequence
The number of terms [tex]\(N\)[/tex] in an arithmetic sequence can be found using the formula:
[tex]\[ N = \frac{l - a}{d} + 1 \][/tex]
Substituting the values:
[tex]\[ N = \frac{(n + 8) - 9}{1} + 1 \][/tex]
[tex]\[ N = \frac{n - 1}{1} + 1 \][/tex]
[tex]\[ N = n - 1 + 1 \][/tex]
[tex]\[ N = n \][/tex]
### Step 3: Apply the sum formula for an arithmetic sequence
The sum [tex]\(S\)[/tex] of the first [tex]\(N\)[/tex] terms of an arithmetic sequence is given by:
[tex]\[ S = \frac{N}{2} \cdot (a + l) \][/tex]
Substituting the values of [tex]\(N\)[/tex], [tex]\(a\)[/tex], and [tex]\(l\)[/tex]:
[tex]\[ S = \frac{n}{2} \cdot (9 + (n + 8)) \][/tex]
[tex]\[ S = \frac{n}{2} \cdot (n + 17) \][/tex]
So, the sum of the arithmetic sequence [tex]\(9, 10, 11, \ldots, n+8\)[/tex] in terms of [tex]\(n\)[/tex] is:
[tex]\[ S = \frac{n(n + 17)}{2} \][/tex]
Thus, the sum of the sequence is:
[tex]\[ \boxed{\frac{n(n + 17)}{2}} \][/tex]
