richie wrote:
Another of my favorites are the first FIVE powers of eleven.
11^0 = 1
11^1 = 11
11^2 = 121
11^3 = 1331
11^4 = 14641
Remember Pascal's Triangle?
I was looking at that for a minute and I realised that there is a simple reason for the pattern:
The binomial theorem states that (a + b)^n = Sum from r=0 to n of nCr a^k b^(n-r) - where nCr is n choose r
letting a=10, b=1 you get 11^n = Sum from r=0 to n of nCr 10^r
The reason the pattern fails at 11^5 is that some of the binomial coefficients (entries in that row of pascal's triangle) have two digits.