Find the sum of all numbers from 150 to 200 which are divisible by 7.

Explanation

To find the sum of all numbers from 150 to 200 that are divisible by 7, we can follow these steps:

### Step 1: Identify the first number divisible by 7 in the range
We start from 150 and check for the first number that is divisible by 7.

- 150 ÷ 7 = 21.428 (not divisible)
- 151 ÷ 7 = 21.571 (not divisible)
- 152 ÷ 7 = 21.714 (not divisible)
- 153 ÷ 7 = 21.857 (not divisible)
- 154 ÷ 7 = 22 (divisible)

**First number divisible by 7 is 154.**

### Step 2: Identify the last number divisible by 7 in the range
Next, we check for the last number that is divisible by 7 before reaching 200.

- 200 ÷ 7 = 28.571 (not divisible)
- 199 ÷ 7 = 28.428 (not divisible)
- 198 ÷ 7 = 28.285 (not divisible)
- 197 ÷ 7 = 28.142 (not divisible)
- 196 ÷ 7 = 28 (divisible)

**Last number divisible by 7 is 196.**

### Step 3: List all the numbers divisible by 7 from 154 to 196
Now we can list the numbers that are divisible by 7 between 154 and 196:

- 154
- 161 (154 + 7)
- 168 (161 + 7)
- 175 (168 + 7)
- 182 (175 + 7)
- 189 (182 + 7)
- 196 (189 + 7)

**The numbers are: 154, 161, 168, 175, 182, 189, 196.**

### Step 4: Count the number of terms
We can see that there are 7 terms in this sequence.

### Step 5: Calculate the sum of the numbers
To find the sum of an arithmetic progression (AP), we can use the formula:

\[ S_n = \frac{n}{2} \times (a + l) \]

Where:
- \( S_n \) = sum of the first n terms
- \( n \) = number of terms
- \( a \) = first term
- \( l \) = last term

Substituting the values we have:
- \( n = 7 \)
- \( a = 154 \)
- \( l = 196 \)

\[ S_7 = \frac{7}{2} \times (154 + 196) \]
\[ S_7 = \frac{7}{2} \times 350 \]
\[ S_7 = \frac{7 \times 350}{2} = \frac{2450}{2} = 1225 \]

**The sum of all numbers from 150 to 200 that are divisible by 7 is 1225.**