Question

If you sum the entries in each row of Pascal's triangle, a pattern emerges. Find a formula that generalizes the result for any row of the triangle, and use it to find the sum of the entries in the 12 th row of the triangle.

Name: Problem 50, Chapter 10: Precalculus
Uploaded: 2020-05-25T16:23:55-08:00
Duration: 2 min 8 s
Channel: Charles Carter
Description: If you sum the entries in each row of Pascal's triangle, a pattern emerges. Find a formula that generalizes the result for any row of the triangle, and use it to find the sum of the entries in the 12 th row of the triangle.

   If you sum the entries in each row of Pascal's triangle, a pattern emerges. Find a formula that generalizes the result for any row of the triangle, and use it to find the sum of the entries in the 12 th row of the triangle.

Precalculus

John W. Coburn 2nd Edition

Chapter 10, Problem 50

Instant Answer

Solved by Expert Charles Carter

05/25/2020

Step 1

The first few rows of Pascal's triangle are as follows: \[ \begin{array}{ccccccccccc} & & & & & 1 & & & & & \\ & & & & 1 & & 1 & & & & \\ & & & 1 & & 2 & & 1 & & & \\ & & 1 & & 3 & & 3 & & 1 & & \\ & 1 & & 4 & & 6 & & 4 & & 1 & \\ \end{array} \] Show more…

Show all steps

Thanks for your feedback!

If you sum the entries in each row of Pascal's triangle, a pattern emerges. Find a formula that generalizes the result for any row of the triangle, and use it to find the sum of the entries in the 12 th row of the triangle.

Play audio

Feedback

Charles Carter and 101 other subject Precalculus educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Exponential Growth in Row Sums

The observation that each row's sum in Pascal's Triangle equals 2^n demonstrates an exponential pattern. This occurs because the expansion of (1+1)^n yields the sum of the binomial coefficients, representing the rapid increase in total as the row number increases.

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two directly above it. It represents binomial coefficients and is fundamental in combinatorial mathematics, illustrating how coefficients appear in binomial expansions.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions of the form (a+b)^n into a sum involving binomial coefficients. By setting both a and b to 1, the theorem shows that the sum of the coefficients in the nth row of Pascal's Triangle is (1+1)^n, which simplifies to 2^n.

Please give Ace some feedback