How many multiples of 3 are there between 100 and 1,000?
Hint: an = a1 + d(n − 1), where a1 is the first term and d is the common difference.
266
299
300
301
The Correct Answer and Explanation is :
To find how many multiples of 3 exist between 100 and 1,000, we first determine the smallest and largest multiples of 3 within this range.
- Finding the First Multiple of 3 Greater Than or Equal to 100:
- Divide 100 by 3: ( \frac{100}{3} \approx 33.33 ).
- The smallest integer greater than 33.33 is 34.
- Thus, the first multiple of 3 is ( 3 \times 34 = 102 ).
- Finding the Last Multiple of 3 Less Than or Equal to 1,000:
- Divide 1,000 by 3: ( \frac{1000}{3} \approx 333.33 ).
- The largest integer less than or equal to 333.33 is 333.
- Therefore, the last multiple of 3 is ( 3 \times 333 = 999 ).
- Identifying the Sequence:
- The multiples of 3 from 102 to 999 form an arithmetic sequence where:
- ( a_1 = 102 ) (the first term),
- ( a_n = 999 ) (the last term),
- ( d = 3 ) (the common difference).
- Using the Formula for the n-th Term:
The formula for the n-th term of an arithmetic sequence is:
[
a_n = a_1 + d(n – 1)
]
Setting ( a_n = 999 ), we substitute:
[
999 = 102 + 3(n – 1)
]
Rearranging gives:
[
999 – 102 = 3(n – 1) \
897 = 3(n – 1) \
n – 1 = \frac{897}{3} = 299 \
n = 300
]
Thus, there are 300 multiples of 3 between 100 and 1,000.
Conclusion
The arithmetic sequence from 102 to 999 with a common difference of 3 yields a total of 300 terms, confirming that the answer is indeed 300. This approach showcases the use of basic arithmetic concepts and sequences to solve problems efficiently.