Question
Describe the pattern formed by the sums of the numbers along the diagonal segments of Pascals Triangle (see figure).
Step 1
It is a triangular array of the binomial coefficients. It starts with a single '1' at the top. Each following row is obtained by adding the number above and to the left with the number above and to the right, treating blank entries as 0. Show more…
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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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