# Formula for the n-th row of Pascal's Triangle

• stgermaine
In summary: What is the sum of the 2nd row?In summary, the formula for the sum of the elements in the nth row of Pascal's Triangle can be found using the binomial theorem and is equal to 2^n. This can be proven through induction by first finding the sum of the 1st and 2nd rows.
stgermaine

## Homework Statement

Find a formula for the sum of the elements of the nth row of Pascal's Triangle

## Homework Equations

C(n,r) = [C(n-1,r-1) + C(n-1,r)]
C(n,0) = C(n,n) = 1

## The Attempt at a Solution

I started with the summation of the elements in the rows n
$\sum^{n}_{r=0} C(n,r)$
= $C(n,0) + \sum^{n-1}_{r=1} C(n,r) + C(n,n)$
= $C(n,0) + \sum^{n-1}_{r=1} [C(n-1,r-1) + C(n-1,r)] + C(n,n)$
= $C(n,0) + \sum^{n-1}_{r=1} C(n-1,r-1) + \sum^{n-1}_{r=1} C(n-1,r) + C(n,n)$
= $C(n,0) + \sum^{n-2}_{r=0} C(n-1,r) + \sum^{n-1}_{r=1} C(n-1,r) + C(n,n)$
= $C(n,0) + [C(n-1,0) + \sum^{n-2}_{r=1} C(n-1,r)] + [\sum^{n-2}_{r=1} C(n-1,r) + C(n-1,n-1)] + C(n,n)$

= $2 \cdot C(n-1,0) + \sum^{n-2}_{r=1} C(n-1,r) + \sum^{n-2}_{r=1} C(n-1,r) + 2 \cdot C(n-1,n-1)$ since C(n,0) = 1 = C(n-1,0) and C(n,n) = 1 = C(n-1,n-1)
=$2 \cdot \sum^{n-1}_{r=0} C(n-1,r)$

This gives the intuitively correct response that the sum of the nth row is twice that of the (n-1)th row. How can I go from here to sum = 2^n?

Thank you!

Do you know the binomial theorem? What is a formula for $(x+y)^n$ in terms of $C(n,r)$?

stgermaine said:
This gives the intuitively correct response that the sum of the nth row is twice that of the (n-1)th row. How can I go from here to sum = 2^n?
This should be a simple induction. What is the sum of the 1st row?

## 1. What is Pascal's Triangle?

Pascal's Triangle is a triangular array of numbers named after the French mathematician Blaise Pascal. Each number in the triangle is the sum of the two numbers directly above it, with the first and last numbers in each row being 1.

## 2. What is the formula for the n-th row of Pascal's Triangle?

The formula for the n-th row of Pascal's Triangle is (n-1)C(r-1), where n is the row number and r is the position of the number in that row. This can also be written as nCr or n choose r.

## 3. How do you construct Pascal's Triangle?

To construct Pascal's Triangle, start with a row of 1's. Then, for each subsequent row, add the two numbers directly above and to the left and right of the current position to get the number in that position. Continue this pattern until the desired number of rows is reached.

## 4. How is Pascal's Triangle related to the binomial coefficients?

The numbers in Pascal's Triangle represent the coefficients of the binomial expansion. For example, the coefficients of (a+b)^3 are 1, 3, 3, and 1, which can be found in the 4th row of Pascal's Triangle.

## 5. What are some applications of Pascal's Triangle?

Pascal's Triangle has applications in algebra, combinatorics, and probability. It can be used to find the coefficients in binomial expansions, calculate combinations and permutations, and solve problems involving probability and combinations. It also has connections to other mathematical concepts such as Pascal's Identity and the Fibonacci Sequence.

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