Find the sum `(41+42+43+…+100).`

Explanation

To find the sum \( S = 41 + 42 + 43 + \ldots + 100 \), we can use the formula for the sum of the first \( n \) natural numbers. The steps are as follows:

### Step 1: Calculate the sum from 1 to 100
The formula for the sum of the first \( n \) natural numbers is given by:
\[ S_n = \frac{n(n + 1)}{2} \]
For \( n = 100 \):
\[ S_{100} = \frac{100(100 + 1)}{2} = \frac{100 \times 101}{2} = \frac{10100}{2} = 5050 \]

### Step 2: Calculate the sum from 1 to 40
Now, we need to calculate the sum from 1 to 40 using the same formula:
For \( n = 40 \):
\[ S_{40} = \frac{40(40 + 1)}{2} = \frac{40 \times 41}{2} = \frac{1640}{2} = 820 \]

### Step 3: Subtract the sum from 1 to 40 from the sum from 1 to 100
Now, we can find the sum from 41 to 100 by subtracting the sum from 1 to 40 from the sum from 1 to 100:
\[ S = S_{100} - S_{40} = 5050 - 820 = 4230 \]

### Final Answer:
Thus, the sum \( 41 + 42 + 43 + \ldots + 100 = 4230 \).

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