The sum of the numbers in the $n$ th row of Pascal's Triangle is $2^{n}$.
Added by Edward C.
Step 1
Step 1: In Pascal's Triangle, each number is the sum of the two numbers directly above it in the previous row. Show more…
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Key Concepts
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a) Determine the sum of the numbers in each of the first five rows in Pascal's triangle. b) What is an expression for the sum of the numbers in the ninth row of Pascal's triangle? c) What is a formula for the sum of the numbers in the $n$ th row?
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Part of Pascal's triangle is shown below. The column on the right represents the sums of the numbers in the rows of Pascal's triangle. 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 2 4 8 16 32 64 a) Based on the evidence in the column on the right, make a conjecture about the sum of the numbers in the 10th row. b) Make a conjecture about the sum of any row.
Eric C.
Sums of Binomial Coefficients Add each of the first five rows of Pascal's triangle, as indicated. Do you see a pattern? $$ \begin{array}{c}{1+1=?} \\ {1+2+1=?} \\ {1+3+3+1=?} \\ {1+4+6+4+1=?} \\ {1+5+10+10+5+1=?}\end{array} $$ Based on the pattern you have found, find the sum of the nth row: $$ \left(\begin{array}{l}{n} \\ {0}\end{array}\right)+\left(\begin{array}{l}{n} \\ {1}\end{array}\right)+\left(\begin{array}{l}{n} \\ {2}\end{array}\right)+\cdots+\left(\begin{array}{l}{n} \\ {n}\end{array}\right) $$ Prove your result by expanding $(1+1)^{n}$ using the Binomial Theorem.
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