Find the sum of the first 1001 terms excluding all multiples of 7
terms of what? If you just mean the first 1001 integers, then
find how many multiples of 7 there are. Say, n of them
Then you want
1001*1002/2 - 7n
Why did the number 7 eat the other numbers at the party? Because it was mean and had a voracious appetite! Now, let's find the sum of the first 1001 terms excluding all multiples of 7. To do that, we need to find the sum of the first 1001 terms and then subtract the sum of all multiples of 7 in that range. Let me do the math for you...
The sum of the first 1001 terms is given by the formula:
S = (n/2)(2a + (n-1)d),
where S is the sum, n is the number of terms, a is the first term, and d is the common difference.
Since the first term is 1 and the common difference is 1 (since we are just adding consecutive numbers), we can plug these values into the formula:
S = (1001/2)(2(1) + (1001-1)(1)),
S = (1001/2)(2 + 1000),
S = (1001/2)(1002).
Now, let's find the sum of all multiples of 7 in the range. We can use another formula for this:
Sn = n/2(2a + (n-1)d),
where Sn is the sum of the first n multiples of 7, a is the first multiple of 7, and d is the common difference (which is 7 since we're adding multiples of 7).
The first multiple of 7 in the range is 7 itself, and the last multiple is 1001.
Sn = n/2(2a + (n-1)d),
Sn = (1001/2)(2(7) + (1001-1)(7)),
Sn = (1001/2)(14 + 7(1000)),
Sn = (1001/2)(14 + 7000),
Sn = (1001/2)(7014).
Now, all we have to do is subtract the sum of the multiples of 7 from the sum of the first 1001 terms:
Final Sum = (1001/2)(1002) - (1001/2)(7014),
Final Sum = (1001/2)(1002 - 7014).
Calculating this gives us the sum of the first 1001 terms excluding all multiples of 7. I hope this mathematical clownery didn't make your head spin!
To solve this problem, we will use the formula for the sum of an arithmetic series.
Step 1: Find the number of terms in the sequence.
The first term is 1, and the common difference is 1. The last term can be calculated as follows:
Last term = first term + (number of terms - 1) * common difference
Last term = 1 + (1001 - 1) * 1
Last term = 1001
Step 2: Find the number of multiples of 7 in the sequence.
We know that every 7th term will be divisible by 7. To find the number of multiples of 7, we can divide the last term by 7 (integer division) and round down.
Number of multiples of 7 = (Last term) // 7 = 1001 // 7 = 143
Step 3: Find the sum of all the terms, including multiples of 7.
To find the sum of the terms including multiples of 7, we can use the formula for the sum of an arithmetic series:
Sum = (number of terms/2) * (first term + last term)
Sum = (1001/2) * (1 + 1001)
Sum = 500.5 * 1002
Sum = 501501
Step 4: Find the sum of the multiples of 7.
To find the sum of the multiples of 7, we can use the formula for the sum of an arithmetic series:
Sum of multiples of 7 = (number of terms/2) * (first term + last term)
Sum of multiples of 7 = (143/2) * (7 + (143 * 7))
Sum of multiples of 7 = 71.5 * 1001
Sum of multiples of 7 = 71771.5
Step 5: Find the sum of the first 1001 terms excluding all multiples of 7.
To find the sum excluding multiples of 7, we can subtract the sum of the multiples of 7 from the sum of all the terms:
Sum excluding multiples of 7 = Sum - Sum of multiples of 7
Sum excluding multiples of 7 = 501501 - 71771.5
Sum excluding multiples of 7 = 429729.5
Therefore, the sum of the first 1001 terms excluding all multiples of 7 is 429729.5.
To find the sum of the first 1001 terms excluding all multiples of 7, we can use the concept of arithmetic series.
An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. In this case, the common difference would be 1, as we are adding each subsequent term by 1.
To exclude multiples of 7, we can use the concept of the arithmetic progression formula, taking into account that the nth term should not be a multiple of 7.
The formula to find the sum of the first n terms of an arithmetic series is:
S = (n/2)(a + l)
where S is the sum of the terms, n is the number of terms, a is the first term, and l is the last term.
To find the last term, we need to determine the largest number less than or equal to 1001 that is not a multiple of 7. To do this, we divide 1001 by 7 and take the floor value. So, the largest number that is not a multiple of 7 is 143.
Next, we need to find the sum of all the terms from 1 to 143. To find the sum of these terms, we can use the formula:
S = (n/2)(a + l)
where n is the number of terms, a is the first term, and l is the last term.
In this case, n is 143, a is 1, and l is 143. Plugging the values into the formula, we get:
S = (143/2)(1 + 143)
S = (143/2)(144)
S = 10302
Therefore, the sum of the first 1001 terms, excluding all multiples of 7, is 10302.