Between 300 and 500 inclusive, find the sums of all the integers that are multiples of:
a. 3
you want the sum of the terms of an arithmetic progression, where
a = 300
d = 3
so, what is the last term?
300 + 3n <= 500
3n <= 200
n <= 66
and the last term is thus 498
so the sum of the 67 terms is 67/2 (300 + 498) = ____
To find the sum of all integers that are multiples of 3 between 300 and 500 (inclusive), we will follow these steps:
Step 1: Identify the first and last multiple of 3 within the given range.
The first multiple of 3 within the range is 303, which is the smallest multiple of 3 greater than or equal to 300.
The last multiple of 3 within the range is 498, which is the largest multiple of 3 less than or equal to 500.
Step 2: Find the number of terms in the sequence.
To find the number of terms, we will use the formula:
Number of terms = (Last term - First term) / Common difference + 1
In this case, the first term is 303, the last term is 498, and the common difference is 3.
Number of terms = (498 - 303) / 3 + 1 = 66
Step 3: Find the sum using the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is:
Sum = (Number of terms / 2) * (First term + Last term)
Sum = (66 / 2) * (303 + 498)
Sum = 33 * (801)
Sum = 26,433
Therefore, the sum of all integers that are multiples of 3 between 300 and 500 (inclusive) is 26,433.
To find the sum of all the integers that are multiples of 3 between 300 and 500 inclusive, we can use a formula for the sum of an arithmetic series.
First, we need to determine the first term (a) and the last term (l) of the sequence. The first multiple of 3 in the given range is 300, and the last multiple of 3 is 498 (the largest multiple of 3 that is less than or equal to 500). Therefore, a = 300 and l = 498.
Next, we need to find the common difference (d). In this case, the common difference is 3, as we are dealing with multiples of 3.
We can use the formula for the sum of an arithmetic series:
S = (n/2) * (a + l)
where S is the sum of the series, n is the number of terms, and a and l are the first and last terms, respectively.
To find the number of terms (n), we can use the formula:
n = ((l - a) / d) + 1
Substituting the known values:
n = ((498 - 300) / 3) + 1
n = (198 / 3) + 1
n = 66 + 1
n = 67
Now we can calculate the sum (S):
S = (67/2) * (300 + 498)
S = 33.5 * 798
S = 26,787
Therefore, the sum of all the integers that are multiples of 3 between 300 and 500 inclusive is 26,787.