To find the sum of the even numbers from 26 to 126, we will follow a series of detailed steps:
1. Identify the Range and Even Numbers:
- The starting number is 26, and the ending number is 126.
- We need to consider only the even numbers within this range.
2. List the Even Numbers:
- The sequence of even numbers starting from 26 up to 126 inclusive is:
[tex]\[
26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126
\][/tex]
3. Count the Even Numbers:
- Let's determine the number of even numbers in this range.
- The even numbers form an arithmetic sequence where the common difference [tex]\(d\)[/tex] is 2 (each number is the previous number plus 2).
- The first term [tex]\(a\)[/tex] of the sequence is 26.
- The last term [tex]\(l\)[/tex] of the sequence is 126.
4. Determine the Number of Terms ([tex]\(n\)[/tex]):
- The formula for the nth term of an arithmetic sequence is:
[tex]\[
l = a + (n - 1)d
\][/tex]
- Rearranging to solve for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{l - a}{d} + 1
\][/tex]
- Substituting the values:
[tex]\[
n = \frac{126 - 26}{2} + 1 = \frac{100}{2} + 1 = 50 + 1 = 51
\][/tex]
- Therefore, there are 51 even numbers in this range.
5. Sum the Even Numbers:
- The sum [tex]\( S \)[/tex] of the first [tex]\( n \)[/tex] terms of an arithmetic sequence can be calculated using the formula:
[tex]\[
S = \frac{n}{2} (a + l)
\][/tex]
- Substituting the values:
[tex]\[
S = \frac{51}{2} (26 + 126) = \frac{51}{2} \times 152 = 51 \times 76 = 3876
\][/tex]
Thus, the sum of the even numbers from 26 to 126 is 3876.
