You have learned that the coefficients in the expansion of (x+y)^n yield a number pyramid called Pascal's triangle. This activity explores some of the interesting properties of this famous number pyramid. Pick any row of Pascal's triangle that comes after the first. Starting at the left end of the row, find the sum of the odd-numbered terms.
Added by Leonard M.
Step 1
First, we need to choose a row of Pascal's triangle that comes after the first. Let's choose the fourth row, which is 1 3 3 1. Show more…
Show all steps
Close
Your feedback will help us improve your experience
Federico Castro and 82 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
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.
Sequences and Series
The Binomial Theorem
Sums of Binomial Coefficients Add each of the first five rows of Pascal's triangle, as indicated. Do you see a pattern? $$\begin{array}{c}{1+1=2} \\ {1+2+1=2} \\ {1+3+3+1=9} \\ {1+4+6+4+1=?} \\{1+5+10+10+5+1=?}\end{array}$$ On the basis of the pattern you have found, find the sum of the $n$ th row: $$\left(\begin{array}{l}{n} \\ {0}\end{array}\right)+\left(\begin{array}{l}{n} \\ {1}\end{array}\right)+\left(\begin{array}{l}{n} \\ {2}\end{array}\right)+\cdots+\left(\begin{array}{l}{n} \\ {n}\end{array}\right)$$ Prove your result by expanding $(1+1)^{n}$ using the Binomial Theorem.
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.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD