Question
61. Find the sum of the numbers in each row of the first ten rows of Pascal's Triangle. Do you sce a pattern?
Step 1
Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it in the previous row. The triangle starts with a 1 at the top, and each row starts and ends with a 1. Show more…
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Find the sum of the numbers in each row of the first ten rows of Pascal's Triangle. Do you see a pattern?
Add each of the first five rows of Pascal's triangle, as indicated. Do you see a pattern? $$\begin{aligned} &1+1=?\\ &1+2+1=?\\ &1+3+3+1=?\\ &1+4+6+4+1=?\\ &1+5+10+10+5+1=? \end{aligned}$$ On the basis of the pattern you have found, find the sum of the $n$ th row: $$\left(\begin{array}{l} n \\ 0 \end{array}\right)+\left(\begin{array}{l} n \\ 1 \end{array}\right)+\left(\begin{array}{l} n \\ 2 \end{array}\right)+\cdots+\left(\begin{array}{l} n \\ n \end{array}\right)$$
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The Binomial Theorem
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