Find the sum of all natural numbers from 1 and 100 which are divisble by 4 or 5.

Explanation

To find the sum of all natural numbers from 1 to 100 that are divisible by 4 or 5, we can follow these steps:

### Step 1: Identify the numbers divisible by 4
The natural numbers from 1 to 100 that are divisible by 4 form an arithmetic progression (AP):
- First term (a) = 4
- Common difference (d) = 4
- Last term (l) = 100

To find the number of terms (n) in this sequence, we can use the formula for the nth term of an AP:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 100 = 4 + (n - 1) \cdot 4 \]
\[ 100 - 4 = (n - 1) \cdot 4 \]
\[ 96 = (n - 1) \cdot 4 \]
\[ n - 1 = \frac{96}{4} = 24 \]
\[ n = 24 + 1 = 25 \]

### Step 2: Calculate the sum of numbers divisible by 4
The sum of the first n terms of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we found:
\[ S_4 = \frac{25}{2} \cdot (4 + 100) \]
\[ S_4 = \frac{25}{2} \cdot 104 \]
\[ S_4 = 25 \cdot 52 = 1300 \]

### Step 3: Identify the numbers divisible by 5
The natural numbers from 1 to 100 that are divisible by 5 also form an AP:
- First term (a) = 5
- Common difference (d) = 5
- Last term (l) = 100

Using the nth term formula again:
\[ 100 = 5 + (n - 1) \cdot 5 \]
\[ 100 - 5 = (n - 1) \cdot 5 \]
\[ 95 = (n - 1) \cdot 5 \]
\[ n - 1 = \frac{95}{5} = 19 \]
\[ n = 19 + 1 = 20 \]

### Step 4: Calculate the sum of numbers divisible by 5
Using the sum formula for the AP:
\[ S_5 = \frac{n}{2} \cdot (a + l) \]
Substituting the values:
\[ S_5 = \frac{20}{2} \cdot (5 + 100) \]
\[ S_5 = 10 \cdot 105 = 1050 \]

### Step 5: Identify the numbers divisible by both 4 and 5 (i.e., divisible by 20)
The natural numbers from 1 to 100 that are divisible by 20 also form an AP:
- First term (a) = 20
- Common difference (d) = 20
- Last term (l) = 100

Using the nth term formula:
\[ 100 = 20 + (n - 1) \cdot 20 \]
\[ 100 - 20 = (n - 1) \cdot 20 \]
\[ 80 = (n - 1) \cdot 20 \]
\[ n - 1 = \frac{80}{20} = 4 \]
\[ n = 4 + 1 = 5 \]

### Step 6: Calculate the sum of numbers divisible by 20
Using the sum formula for the AP:
\[ S_{20} = \frac{n}{2} \cdot (a + l) \]
Substituting the values:
\[ S_{20} = \frac{5}{2} \cdot (20 + 100) \]
\[ S_{20} = \frac{5}{2} \cdot 120 \]
\[ S_{20} = 5 \cdot 60 = 300 \]

### Step 7: Calculate the final sum
Now, we can find the total sum of numbers divisible by 4 or 5 using the principle of inclusion-exclusion:
\[ S = S_4 + S_5 - S_{20} \]
Substituting the values:
\[ S = 1300 + 1050 - 300 \]
\[ S = 2050 \]

Thus, the sum of all natural numbers from 1 to 100 that are divisible by 4 or 5 is **2050**.