The given sequence is an arithmetic sequence where the first term \(a = 1\) and the common difference \(d = 5\).

We need to find the sum of this arithmetic sequence from 1 to 100. To do so, we'll use the formula for the sum \(S_n\) of an arithmetic sequence:

\[
S_n = \frac{n}{2} \times (a + l)
\]

Where:
- \(n\) is the number of terms,
- \(a\) is the first term,
- \(l\) is the last term (100 in this case).

### Step 1: Find the number of terms \(n\)
The general term of an arithmetic sequence is given by:
\[
a_n = a + (n-1) \cdot d
\]

We know the last term \(a_n = 100\), so we solve for \(n\):

\[
100 = 1 + (n-1) \cdot 5
\]

\[
99 = (n-1) \cdot 5
\]

\[
n-1 = \frac{99}{5} = 19.8
\]

So, \(n = 20.8 + 1 = 21\)



Learn how to calculate the sum of the arithmetic sequence 1+5+10+...+100 with a step-by-step breakdown. This problem explores arithmetic sequences, series summation, and includes key formulas for calculating the number of terms and the sum. Ideal for students in Grades 9-12.

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