@article {MATHEDUC.05988817, author = {Cox, Roger D.}, title = {Delving deeper: Patterns in binomial coefficients and repeated differences.}, year = {2011}, journal = {Mathematics Teacher}, volume = {104}, number = {6}, issn = {0025-5769}, pages = {470-475}, publisher = {National Council of Teachers of Mathematics (NCTM), Reston, VA}, abstract = {From the introduction: My general classroom introduction to sequences and series led to a discussion of number patterns. One example given was the fact that the sum of the first \$n\$ odd numbers is \$n^2\$. As class ended, a student approached to say that he had noticed that taking repeated differences of consecutive squares always resulted in the constant 2. I pointed out that since summing consecutive odd numbers resulted in consecutive squares, it followed that the difference of consecutive squares would result in consecutive odd numbers. Therefore, the second differences would always be 2. The student found this interesting. As I collected my things, he commented that he had found that repeated differences of other powers seemed to result in other constants. Was this observation always true? If it were true, was there any way to know the power if given the constant? I said I would look into his assertion and question. That night, a few binomial expansions and differences later, the answer seemed clear. The pattern was taking shape. As I followed up with the fourth and fifth powers to confirm, what was happening became clearer. In answering my student\rq s question the next day, I noted that taking \$k+1\$ or more consecutive positive integers to the \$k\$th power and taking the \$k\$th differences resulted in the constant \$k\$!. This result had been demonstrated for \$k=1\$ to \$k=5\$ using binomial expansions. It was strongly suggested for \$k\$ in general and had been confirmed for some special cases.}, msc2010 = {H20xx (K20xx E50xx F60xx)}, identifier = {2012a.00542}, }