Fun with Numbers

On a rainy day I happened to scribble some numbers on a piece of paper and was looking for some connection in those numbers. Suddenly I landed upon a strange connection, a strange occurrence which I want to share with all.

Step 1: Let us write numbers from zero to any number (for example from 0 to 20) under a column x1. Deduct the number in the first row (0) from the number in the second row (1) and write the difference under the second column CD1. Then deduct the number in the second row under column x1 (1) from the number in the third row (2) and write the difference under column CD1. Continue like this and write the difference from consecutive numbers under the column CD1. The numbers under the column CD1 will be 1,1,1,1 etc.













Step 2: Now square the numbers under column x1 and write the squares under column x2. The numbers will be 0,1,4,9 etc. Again deduct the first number under column x2 (0) from the number in the second row (1) and write the difference under column D2. Then deduct the number in the second row under column x2 from the number in the third row and so on and so forth. The numbers under the column D2 will be 1,3,5,7 etc. Now deduct the number in the first row under column D2 from the number in the second row and so on and so forth, and write the differences under column CD2. The numbers under the column CD2 will be 2,2,2,2 etc.











Step 3: Now find the cubes of numbers under the column x1 and write the cubes under column
x3. The numbers will be 0,1,8,27,64 etc. Again deduct the number in the first row under column x3 from the number in the second row and write the difference under column D31. The numbers will be 1, 7, 19, 37 etc. Now deduct the number in the first row under column D31 from the number in the second row and so on so forth and write the differences under column D32. The number will be 6, 12, 18, etc. If you do the same exercise with numbers under column D32 you will get a common difference of 6 which is to be written under column CD3. Please find attached herewith a tabulation in which similar exercise has been done for numbers x4, x5, x6 and x7.

If one observes the numbers under column CD1 (1), CD2 (2), CD3 (6), CD4 (24), CD5 (120), CD6 (720) and CD7 (5040), one can see the common differences following a pattern which can be described by the following equation.

CD_n = CD_(n-1) * n.

Where CD_n = common difference for numbers raised to the power of n

CD_(n-1) = common difference for numbers raised to the power of (n-1)

It can also be observed that when n=1, the common difference occurs in the first column to right of numbers under x¹. When n=2, the common difference occurs in the second column to the right of column x² and when n=3, the common difference occurs in the third column to the right of column x³ and so on. In general the common difference between numbers raised to the power of n occurs at the n^th column to the right of column xⁿ. Obviously all the common differences (2,6,24, 120,720 etc.) are even numbers except the first one (1).