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

Answer 1

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!