# Q.E.D.

54 55 Deep results are not usually discovered ataglance but rather as a result of a process of ever increasing generalisation. In this appendix, we sketch part of such a process involving Pascals Triangle and its relative, the Power Triangle. These general formul work for any choice of a and b. Here is how you derive a general formula for a bn which works for any a and b and natural number n. Pascals triangle on the left summarises the formul on the right. Can you prove that each of its entries except the tip is the sum of the numbers right above it On page 21, we saw that Pascals triangle co incides with the triangle on the left. Here is the number of different ways to choose k objects among n objects, which is 1 if k is zero and otherwise is n.n 1.n 2...n k 11.2.3 ... k. This gives the famous Binomial Theorem aPPendIX Iv trIangles oF generalIty Many generalities hide in Pascals triangle. For example, on page 43 we saw that its nth diagonal adds up to the nth Fibonacci number f n . We express this as follows The structure of Pascals triangle quickly suggests the Golf Club Theorem Highlight a golfclubshaped pattern of numbers in the triangle as in the examples on the left. The numbers in the handle of the club then add up to the one in its tip Golf clubs with handles along the outside first, white column on the left translate into the formula Golf clubs with handles in the second column translate into the formula for the sum of the first n natural numbers see also pages 32 and 34 These sums are called triangular numbers because they count the numbers of circles in the triangular patterns on the left. Similarly, it follows that the numbers in the next column are pyramidal numbers. In general, the numbers in column n of Pascals triangle are the n1 dimensional pyramidal numbers. The Power Triangle below was only discovered recently. It summarises the general formul on the right below, some of which we proved on page 34. This triangle grows just like Pascals except that you have to multiply a number by its suffix before adding it. For example, in the fourth row 7.2 12.3 50 and 12.3 6.4 60.