Let's tackle the given problems step by step.
### Part (a): [tex]\(1 + 2 + 3 + \cdots + 100\)[/tex]
Observe the given pattern, which shows that the sum of the first [tex]\( n \)[/tex] natural numbers can be expressed as:
[tex]\[1 + 2 + 3 + \cdots + n = \frac{n \times (n+1)}{2}\][/tex]
For [tex]\( n = 100 \)[/tex],
[tex]\[
\text{Sum} = \frac{100 \times 101}{2}
\][/tex]
[tex]\[
\text{Sum} = \frac{10100}{2}
\][/tex]
[tex]\[
\text{Sum} = 5050
\][/tex]
Therefore, the value of [tex]\(1 + 2 + 3 + \cdots + 100\)[/tex] is [tex]\(5050\)[/tex].
### Part (b): [tex]\(21 + 22 + 23 + \cdots + 50\)[/tex]
To find the sum of an arithmetic series, we use the following formula:
[tex]\[
\text{Sum} = \frac{n}{2} \times (\text{first term} + \text{last term})
\][/tex]
Where [tex]\( n \)[/tex] is the number of terms in the series.
1. First, find the number of terms ([tex]\( n \)[/tex]):
[tex]\(
n = ( \text{last term} - \text{first term} ) + 1
\)[/tex]
For the series [tex]\(21, 22, 23, \cdots, 50\)[/tex]:
[tex]\[
n = (50 - 21) + 1
\][/tex]
[tex]\[
n = 30
\][/tex]
2. Calculate the sum using the arithmetic series formula:
[tex]\(
\text{first term} = 21 \\
\text{last term} = 50
\)[/tex]
[tex]\[
\text{Sum} = \frac{30}{2} \times (21 + 50)
\][/tex]
[tex]\[
\text{Sum} = 15 \times 71
\][/tex]
[tex]\[
\text{Sum} = 1065
\][/tex]
Therefore, the value of [tex]\(21 + 22 + 23 + \cdots + 50\)[/tex] is [tex]\(1065\)[/tex].
### Final Results:
(a) The value of [tex]\(1 + 2 + 3 + \cdots + 100\)[/tex] is [tex]\(5050\)[/tex].
(b) The value of [tex]\(21 + 22 + 23 + \cdots + 50\)[/tex] is [tex]\(1065\)[/tex].
