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Method 1 (Arithmetic Sequence): The sum of the first 1,000,000 positive even numbers can be expressed in an arithmetic sequence: 2 + 4 + 6 + 8 + ... + 1,999,998 + 2,000,000 The sum of the first n terms of an arithmetic sequence is given in the equation: n(a1 + an)/2 Where n is the number of terms (in this case 1,000,000), a1 is the first term (in this case 2), and an is the nth term (in this case the 1,000,000th term which is 2,000,000). So the sum is 1,000,000(2 + 2,000,000)/2 = 1,000,001,000,000. Method 2 (Algebraic Solution): The positive even numbers are {2, 4, 6, 8, 10, ...}. The sum of the first 1,000,000 of these numbers can be expressed as: (2) + (4) + (6) + ... + (2,000,000) Note that each term n can be expressed as 2 + 2(n-1). So the last term is 2 + 2(1,000,000 - 1) = 2,000,000. Now for some tricky equations. We could simply add this out, but that would take much longer than we would like. We'll start by letting x equal the sum of the first 1,000,000 positive even numbers (the value shown above). So: x = 2 + 4 + ... + 1,999,998 + 2,000,000 x = 2,000,000 + 1,999,998 + ... + 4 + 2 Note that the second equation is simply the first one reversed which we can do because addition is associative (meaning we can add in any order we want). Now we will add the equations together (which we can do similar to elimination of multi-variable equations). We start by adding the first terms, then the second terms, and so on. 2x = 2,000,002 + 2,000,002 + ... + 2,000,002 + 2,000,002 Well, how many 2,000,002s are there? Well, 1,000,000 (we were adding the first 1,000,000 positive even numbers). So now we can simplify this expression by saying: 2x = 1,000,000 * 2,000,002 Now we solve by multiplying the numbers on the right and dividing by two (to get the 2x to be an x). 2x = 2,000,002,000,000 x = 1,000,001,000,000 So the sum of the first 1,000,000 positive even numbers is 1,000,001,000,000!

Q: What is the sum of the first 1000000 positive even numbers?

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The sum of the first 1,000,000 positive even numbers is: 2 + 4 + 6 + 8 + ... + 2,000,000 The sum of the first 1,000,000 positive odd integers is: 1 + 3 + 5 + 7 + ... + 1,999,999 The difference between the two is: (2-1) + (4-3) + (6-5) + (8-7) + ... + (2,000,000-1,999,999). This is the same as: 1 + 1 + 1 + 1 + ... + 1. Well how many 1's are there? 1,000,000. So the difference is 1,000,000. Note that if the question asked for the difference between the sum of the first 1,000 positive even numbers and the sum of the first 1,000 positive odd numbers, the answer would be 1,000. The first n even numbers and odd numbers? n.

The first 100 even numbers are: 2 468101214161820222426283032343638404244464850525456586062646668707274767880828486889092949698100102104106108110112114116118120122124126128130132134136138140142144146148150152154156158160162164166168170172174176178180182184186188190192194196198200.

They are called integers. Fractions and decimals are NOT integers, even if it has a positive or negative sign.

The numbers are 12 and 14.

Composite numbers are numbers composed of 2 or more primes. 2 is a prime number. Any other (positive) even number is a product of 2 and some other number (not 1).

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The sum of the first 1,000,000 positive even numbers is: 2 + 4 + 6 + 8 + ... + 2,000,000 The sum of the first 1,000,000 positive odd integers is: 1 + 3 + 5 + 7 + ... + 1,999,999 The difference between the two is: (2-1) + (4-3) + (6-5) + (8-7) + ... + (2,000,000-1,999,999). This is the same as: 1 + 1 + 1 + 1 + ... + 1. Well how many 1's are there? 1,000,000. So the difference is 1,000,000. Note that if the question asked for the difference between the sum of the first 1,000 positive even numbers and the sum of the first 1,000 positive odd numbers, the answer would be 1,000. The first n even numbers and odd numbers? n.

The sum of the first 30 positive even numbers is 930.

The sum of the first 100 positive even numbers is 10,100.

The sum of the first 30 positive even numbers is 930.

The sum of the first 100 positive even numbers is 10,100.

twenty

The answer to what is the sum of the first positive even numbers is 930. It doesn't mean that you add all the even numbers up until you reach the number 30. In fact, since you have to get the sum of the even numbers, you have to add all even numbers until you get to the number 60.

There is no conjecture about the sum of the first 30 positive even numbers. The answer can be derived and proven. A statement that has been proven is no longer a conjecture.

About half the total sum listed.

the first positive even numbers are 2,4,6,8,10

The sum of the first 15 positive even numbers is 240. (Simply square 15, then add 15 to the result: 15 x 15 = 225. 225 + 15 = 240).

There is only one even prime number...2