Subject: Sums of Evens How do I find a geometric way to easily compute sums of consecutive even numbers 2 + 4 + 6 + .... Hi Rosa, The first part of our answer to your question comes from the notes for an online class here at the University of Regina. The problem was to find the sum 1 + 2 + 3 + ... + 5000. The solution begins:
Start with a smaller problem:
Carry out the PlanFirst, we'll show how to sum 1 + 2 + 3 + 4 + 5 as follows. If we count the blocks in the array, we will see that the sum is 15.
The creative, new approach is a famous technique attributed to Carl Friedrich Gauss (1777 - 1855). Gauss amazed his teachers in elementary school by seeing quickly how to sum the integers from 1 to 100.
Be creativeIf we have two copies of the block array above we can flip one over and stack them to create a rectangle. This rectangle is 5 blocks wide and 6 blocks high. The number of blocks in the rectangular array is
+ (5 + 4 + 3 + 2 + 1) Which is equivalent to 1 + 2 + 3 + 4 + 5 = ^{30}/_{2} = 15
The rectangular array is five columns of 6 blocks each. The same structure can be seen in the numeric array + (5 + 4 + 3 + 2 + 1) where there are 5 columns of numbers and the sum of each column is 6. The first n natural numbers can be derived using this numeric array: Write out the sum forwards:
Note: there are n terms being summed, each of which is (n+1) so the sum of the forwards and backwards together is n(n+1). But that is twice the sum of 1 to n since we added it up forwards and backwards so to get the sm of 1 to n, we need to divide by two. Hence the formula is A similar approach can be used to find the sum of the first n even numbers. Use a geometric array of blocks as in the diagrams above or a numeric array obtained by writing the sum forward and backwards and then adding the columns. Leeanne and HarleyTo return to the previous page use your browser's back button. |