Solution:
Given:
Integers from 1 to 100.
Integers are whole numbers that can be positive, negative or zero.
Hence, integers between 1 to 100 are positive whole numbers between 1 and 100.
This is an example of arithmetic sequence increasing by a common difference of 1.
Using the formula for the sum of an arithmetic sequence,
[tex]\begin{gathered} S_n=\frac{n}{2}(2a+(n-1)d) \\ \text{where;} \\ n\text{ is the number of terms, n = 100} \\ a\text{ is the first term, a = 1} \\ d\text{ is the co}mmon\text{ difference, d = 1} \end{gathered}[/tex]Hence,
[tex]\begin{gathered} S_n=\frac{n}{2}(2a+(n-1)d) \\ S_{100}=\frac{100}{2}(2(1)+(100-1))1 \\ S_{100}=50(2+99) \\ S_{100}=50(101) \\ S_{100}=50\times101 \\ S_{100}=5050 \end{gathered}[/tex]Alternatively using another formula,
[tex]\begin{gathered} S_n=\frac{n}{2}(a+l) \\ \text{where;} \\ a\text{ is the first term, a = 1} \\ n\text{ is the number of terms, n = 100} \\ l\text{ is the last term, l = 100} \\ \\ S_n=\frac{n}{2}(a+l) \\ S_{100}=\frac{100}{2}(1+100) \\ S_{100}=50(101) \\ S_{100}=50\times101 \\ S_{100}=5050 \end{gathered}[/tex]Therefore, the sum of the integers from 1 to 100 is 5050.