Chapter 1.2, Problem 29ES

Pascal’s Triangle The triangular pattern in the following figure is known as Pascal’s triangle. Pascal's triangle has intrigued mathematician for hundreds of years. Although it is named after the mathematician Blaise Pascal (1623-1662), there is evidence that it was first developed in China in the 1300s. The numbers in Pascal's triangle are created in the following manner. Each row begins and ends with the number 1. Any other number in a row is the sum of the two closest numbers about it. For instance, the first 10 in raw 5 is the sum of the first 4 and the 6 above it in raw 4.

There are many patterns that can be discovered in Pascal's triangle.

a. Find the sum of the numbers in each row, except row 0, of the portion of Pascal's triangle shown above. What pattern do you observe concerning mesa sums? Predict the sum of the numbers in row 9 of Pascal’s triangle.

b. The numbers $1,3,6,10,15,\mathrm{...},\frac{n\left(n+1\right)}{2},\mathrm{...}$ are called triangular number. Where do the triangular number appear in Pascal’s triangle?