Chapter 6.1, Problem 37E

Solution

To describe: The pattern formed by the sums of the numbers along the diagonal segment of Pascal’s Triangle.

Fibonacci Pattern

Given information:

The Pascal Triangle is given as

  

Calculation:

Pascal’s Triangles for combination is shown below.

  

Pascal’s Triangles as numbers can be written as

  $\begin{array}{ccccccccccccccc}& & & & & & & 1& & & & & & & \\ & & & & & & 1& & 1& & & & & & \\ & & & & & 1& & 2& & 1& & & & & \\ & & & & 1& & 3& & 3& & 1& & & & \\ & & & 1& & 4& & 6& & 4& & 1& & & \\ & & 1& & 5& & 10& & 10& & 5& & 1& & \\ & 1& & 6& & 15& & 20& & 15& & 6& & 1& \\ 1& & 7& & 21& & 105& & 105& & 21& & 7& & 1\end{array}$

Every numbers in Pascal’s Triangle except 1 is some of the closest two numbers in row directly above it.

Along diagonal of the Pascal’s Triangle each number is sum of the numbers of the above closest diagonal.

So, the pattern is

  $\begin{array}{l}1=1\\ 1+1=2\\ 1+2=3\\ 3+1+1=5=3+2\\ 1+4+3=8=5+3\end{array}$

So, the numbers will be 1, 2, 3, 5, 8, 13,…which will form a Fibonacci series.