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Answer :
To solve the problem of finding the sum of all natural numbers between 250 and 1000 that are divisible by 3, we can follow these simple steps:
1. Identify the Range and Conditions: We want numbers between 250 and 1000 that are divisible by 3.
2. Find the First Number Divisible by 3 Greater than 250:
- The first number greater than 250 that is divisible by 3 is 252.
3. Find the Last Number Divisible by 3 Less than 1000:
- The last number less than 1000 that is divisible by 3 is 999.
4. Determine the Sequence of Numbers:
- The numbers that satisfy our condition form an arithmetic sequence with the first term (a) as 252 and the last term (l) as 999. The common difference (d) is 3 because we are considering numbers divisible by 3.
5. Calculate the Number of Terms in the Sequence:
- The formula for the number of terms in an arithmetic sequence is:
[tex]\[
n = \frac{{l - a}}{d} + 1
\][/tex]
- Plugging in the values:
[tex]\[
n = \frac{{999 - 252}}{3} + 1 = 250
\][/tex]
6. Calculate the Sum of the Sequence:
- Use the formula for the sum of an arithmetic sequence:
[tex]\[
\text{Sum} = \frac{n}{2} \times (a + l)
\][/tex]
- Plugging in the values:
[tex]\[
\text{Sum} = \frac{250}{2} \times (252 + 999) = 125 \times 1251 = 156,375
\][/tex]
Therefore, the sum of all natural numbers between 250 and 1000 that are exactly divisible by 3 is 156,375. The correct answer is option (d) 156,375.
1. Identify the Range and Conditions: We want numbers between 250 and 1000 that are divisible by 3.
2. Find the First Number Divisible by 3 Greater than 250:
- The first number greater than 250 that is divisible by 3 is 252.
3. Find the Last Number Divisible by 3 Less than 1000:
- The last number less than 1000 that is divisible by 3 is 999.
4. Determine the Sequence of Numbers:
- The numbers that satisfy our condition form an arithmetic sequence with the first term (a) as 252 and the last term (l) as 999. The common difference (d) is 3 because we are considering numbers divisible by 3.
5. Calculate the Number of Terms in the Sequence:
- The formula for the number of terms in an arithmetic sequence is:
[tex]\[
n = \frac{{l - a}}{d} + 1
\][/tex]
- Plugging in the values:
[tex]\[
n = \frac{{999 - 252}}{3} + 1 = 250
\][/tex]
6. Calculate the Sum of the Sequence:
- Use the formula for the sum of an arithmetic sequence:
[tex]\[
\text{Sum} = \frac{n}{2} \times (a + l)
\][/tex]
- Plugging in the values:
[tex]\[
\text{Sum} = \frac{250}{2} \times (252 + 999) = 125 \times 1251 = 156,375
\][/tex]
Therefore, the sum of all natural numbers between 250 and 1000 that are exactly divisible by 3 is 156,375. The correct answer is option (d) 156,375.
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