Let's solve the problem step by step based on the given pattern and information.
Part (a): Finding the sum [tex]\(1 + 2 + 3 + \ldots + 100\)[/tex]
We observe the pattern in the problem:
[tex]\[ 1 = \frac{1 \times 2}{2} \][/tex]
[tex]\[ 1 + 2 = \frac{2 \times 3}{2} \][/tex]
[tex]\[ 1 + 2 + 3 = \frac{3 \times 4}{2} \][/tex]
[tex]\[ 1 + 2 + 3 + 4 = \frac{4 \times 5}{2} \][/tex]
Based on these examples, we see that the sum of the first [tex]\(n\)[/tex] natural numbers can be written as:
[tex]\[ 1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2} \][/tex]
Here, [tex]\(n = 100\)[/tex]. Substituting 100 for [tex]\(n\)[/tex], we get:
[tex]\[ 1 + 2 + 3 + \ldots + 100 = \frac{100 \times 101}{2} \][/tex]
Thus,
[tex]\[ 1 + 2 + 3 + \ldots + 100 = \frac{100 \times 101}{2} = 5050 \][/tex]
So, the value of [tex]\(1 + 2 + 3 + \ldots + 100\)[/tex] is 5050.
Part (b): Finding the sum [tex]\(21 + 22 + 23 + \ldots + 50\)[/tex]
To find the sum of any arithmetic progression, we use the formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic progression:
[tex]\[ S_n = \frac{n}{2}(a + l) \][/tex]
where:
- [tex]\(S_n\)[/tex] is the sum of the arithmetic sequence,
- [tex]\(n\)[/tex] is the number of terms,
- [tex]\(a\)[/tex] is the first term,
- [tex]\(l\)[/tex] is the last term.
In this case, the first term [tex]\(a\)[/tex] is 21, and the last term [tex]\(l\)[/tex] is 50. To find the number of terms ([tex]\(n\)[/tex]), we calculate the difference between the last and first terms and add 1:
[tex]\[ n = 50 - 21 + 1 = 30 \][/tex]
Now, substituting the values into the sum formula:
[tex]\[ S_{30} = \frac{30}{2} \times (21 + 50) \][/tex]
[tex]\[ S_{30} = 15 \times 71 \][/tex]
[tex]\[ S_{30} = 1065 \][/tex]
So, the value of [tex]\(21 + 22 + 23 + \ldots + 50\)[/tex] is 1065.
Summary:
- The value of [tex]\( 1 + 2 + 3 + \ldots + 100 \)[/tex] is 5050.
- The value of [tex]\( 21 + 22 + 23 + \ldots + 50 \)[/tex] is 1065.
