Let's analyze and solve each part of the problem step-by-step.
### Part (a): Sum of the first 100 natural numbers
We observe the given pattern where the sum of the first [tex]\( n \)[/tex] natural numbers is represented as:
[tex]\[ 1 + 2 + 3 + \ldots + n = \frac{n \times (n+1)}{2} \][/tex]
To find the sum of the first 100 natural numbers, we set [tex]\( n = 100 \)[/tex]:
[tex]\[ 1 + 2 + 3 + \ldots + 100 = \frac{100 \times 101}{2} = 5050 \][/tex]
Therefore, the value of the sum [tex]\( 1 + 2 + 3 + \ldots + 100 \)[/tex] is:
[tex]\[ 5050 \][/tex]
### Part (b): Sum of the numbers from 21 to 50
For this, we use the formula for the sum of an arithmetic series. The sum [tex]\( S \)[/tex] of an arithmetic series is given by:
[tex]\[ S = \frac{n}{2} \times (\text{first term} + \text{last term}) \][/tex]
where [tex]\( n \)[/tex] is the number of terms in the series.
We need to find the sum of the numbers from 21 to 50. Let's identify the necessary components first:
- First term (a) = 21
- Last term (l) = 50
The number of terms [tex]\( n \)[/tex] can be calculated as:
[tex]\[ n = 50 - 21 + 1 = 30 \][/tex]
Now we can apply the formula:
[tex]\[ S = \frac{30}{2} \times (21 + 50) = 15 \times 71 = 1065 \][/tex]
Therefore, the value of the sum [tex]\( 21 + 22 + 23 + \ldots + 50 \)[/tex] is:
[tex]\[ 1065 \][/tex]
### Final Answers:
(a) The value of [tex]\( 1 + 2 + 3 + \ldots + 100 \)[/tex] is:
[tex]\[ 5050 \][/tex]
(b) The value of [tex]\( 21 + 22 + 23 + \ldots + 50 \)[/tex] is:
[tex]\[ 1065 \][/tex]
