What is the difference between the sum of the first 400 even counting numbers and the sum of the first 400 odd counting numbers?
To find the difference between the sum of the first 400 even counting numbers and the sum of the first 400 odd counting numbers, we first need to determine the formulas to compute these sums.
The sum of the first n even counting numbers can be calculated using the formula:
Sum of even numbers = n * (n + 1)
Similarly, the sum of the first n odd counting numbers is calculated using the formula:
Sum of odd numbers = n^2
Now, let's apply these formulas to find the difference.
For the sum of the first 400 even counting numbers:
Sum of even numbers = 400 * (400 + 1) = 400 * 401
For the sum of the first 400 odd counting numbers:
Sum of odd numbers = 400^2 = 160,000
To find the difference, we subtract the sum of the first 400 odd numbers from the sum of the first 400 even numbers:
Difference = Sum of even numbers - Sum of odd numbers
Difference = 400 * 401 - 160,000
Now we can calculate the difference using a calculator or by performing the arithmetic:
Difference = 160,400 - 160,000
Difference = 400
Therefore, the difference between the sum of the first 400 even counting numbers and the sum of the first 400 odd counting numbers is 400.
To find the difference between the sum of the first 400 even counting numbers and the sum of the first 400 odd counting numbers, we need to calculate each sum separately.
1. Sum of the first 400 even counting numbers:
The even counting numbers can be represented by the sequence of multiples of 2: 2, 4, 6, 8, ...
The general formula for the nth term of this sequence is: 2n.
To find the sum of the sequence, we use the formula for the sum of an arithmetic series:
Sn = (n/2)(a + l)
where Sn represents the sum of n terms, a is the first term, and l is the last term.
In this case, we want the sum of the first 400 even counting numbers, so n = 400, a = 2, and l = 2(400) = 800.
Plugging these values into the formula, we get:
Sum_even = (400/2)(2 + 800)
= 200(802)
= 160,400.
2. Sum of the first 400 odd counting numbers:
The odd counting numbers can be represented by the sequence of multiples of 2 with 1 subtracted: 1, 3, 5, 7, ...
The general formula for the nth term of this sequence is: 2n - 1.
Using the same formula for the sum of an arithmetic series, we can find the sum of the first 400 odd counting numbers.
In this case, n = 400, a = 1, and l = 2(400) - 1 = 799.
Plugging these values into the formula, we get:
Sum_odd = (400/2)(1 + 799)
= 200(800)
= 160,000.
Therefore, the difference between the sum of the first 400 even counting numbers and the sum of the first 400 odd counting numbers is:
Sum_even - Sum_odd = 160,400 - 160,000 = 400.