Final answer:
Calculate the sum of all multiples of 3 between 1 and 1000 by identifying the first and last multiples within the range, determining the number of terms, and using the arithmetic series sum formula to get 166833.
Step-by-step explanation:
The problem at hand is to calculate the sum of all the multiples of 3 that lie between 1 and 1000. To solve this, we should first identify the smallest and the largest multiple of 3 within this range. The smallest multiple is 3 itself, and the largest is 999, as it is the largest number less than 1000 that is divisible by 3. Now, notice that this is an arithmetic sequence where each term increases by 3. We can use the formula for the sum of an arithmetic series, which is S = n/2 × (a1 + an), where n is the number of terms, a1 is the first term, and an is the last term.
To find n, the number of terms, we divide the largest multiple by 3 and then subtract 1 (for the initial 3), and add back 1 to include both ends of the series. This gives us:
n = (999/3) - 1 + 1 = 333
Knowing n, we now plug all values into the sum formula:
S = (333/2) × (3 + 999) = 166.5 × 1002 = 166833
Therefore, the sum of all multiples of 3 between 1 and 1000 is 166833.