To solve the problem of finding the sum of the arithmetic sequence [tex]\( 3 + 8 + 13 + \cdots + 293 + 298 + 303 \)[/tex], we can follow a detailed, step-by-step approach as follows:
### Step-by-Step Solution:
1. Identify the Arithmetic Sequence Parameters:
- First term [tex]\( (a) \)[/tex]: The sequence starts at [tex]\(3\)[/tex], so [tex]\(a = 3\)[/tex].
- Common difference [tex]\( (d) \)[/tex]: Each term increases by [tex]\(5\)[/tex], so [tex]\(d = 5\)[/tex].
- Last term [tex]\( (l) \)[/tex]: The sequence ends at [tex]\(303\)[/tex], so [tex]\(l = 303\)[/tex].
2. Find the Number of Terms [tex]\( (n) \)[/tex]:
We use the formula for the nth term of an arithmetic sequence:
[tex]\[
a_n = a + (n-1)d
\][/tex]
Here, [tex]\(a_n = 303\)[/tex], [tex]\(a = 3\)[/tex], and [tex]\(d = 5\)[/tex]. Solve for [tex]\(n\)[/tex]:
[tex]\[
303 = 3 + (n-1) \cdot 5
\][/tex]
Simplify and solve for [tex]\(n\)[/tex]:
[tex]\[
303 - 3 = (n-1) \cdot 5 \\
300 = (n-1) \cdot 5 \\
60 = n-1 \\
n = 61
\][/tex]
So, there are [tex]\(61\)[/tex] terms in the sequence.
3. Calculate the Sum of the Sequence:
The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is given by the formula:
[tex]\[
S_n = \frac{n}{2} \cdot (a + l)
\][/tex]
Substitute [tex]\(n = 61\)[/tex], [tex]\(a = 3\)[/tex], and [tex]\(l = 303\)[/tex] into the formula:
[tex]\[
S_{61} = \frac{61}{2} \cdot (3 + 303)
\][/tex]
Simplify inside the parentheses:
[tex]\[
S_{61} = \frac{61}{2} \cdot 306 \\
S_{61} = 30.5 \cdot 306 \\
S_{61} = 9333.0
\][/tex]
Thus, the sum of the arithmetic sequence [tex]\( 3 + 8 + 13 + \cdots + 293 + 298 + 303 \)[/tex] is [tex]\( \boxed{9333.0} \)[/tex].
