Prove that for any positive integer n ( n 0 )-( n 1

Name: Problem 60, Chapter 12: Contemporary Precalculus
Uploaded: 2020-07-17T18:32:15-08:00
Duration: 1 min 2 s
Channel: Linh Vu
Description: Prove that for any positive integer n ( n 0 )-( n 1 )+( n 2 )-( n 3 )+( n 4 )-⋯ +(-1)^k( n k )+⋯+(-1)^n( n n )=0

step 1 Here if we can see the special case for the 1 minus 1 power n using the binomial theorem which w

Question

Prove that for any positive integer $n$ $$\begin{array}{l}\left(\begin{array}{l}n \\0\end{array}\right)-\left(\begin{array}{l}n \\1\end{array}\right)+\left(\begin{array}{l}n \\ 2\end{array}\right)-\left(\begin{array}{l}n \\3\end{array}\right)+\left(\begin{array}{l}n \\4\end{array}\right)-\cdots \\+(-1)^{k}\left(\begin{array}{l}n \\k\end{array}\right)+\cdots+(-1)^{n}\left(\begin{array}{l}n \\n\end{array}\right)=0\end{array}$$

   Prove that for any positive integer $n$
$$\begin{array}{l}\left(\begin{array}{l}n \\0\end{array}\right)-\left(\begin{array}{l}n \\1\end{array}\right)+\left(\begin{array}{l}n \\
2\end{array}\right)-\left(\begin{array}{l}n \\3\end{array}\right)+\left(\begin{array}{l}n \\4\end{array}\right)-\cdots \\+(-1)^{k}\left(\begin{array}{l}n \\k\end{array}\right)+\cdots+(-1)^{n}\left(\begin{array}{l}n \\n\end{array}\right)=0\end{array}$$

Contemporary Precalculus

Thomas W.… 5th Edition

Chapter 12, Problem 60

Instant Answer

Solved by Expert Linh Vu

07/17/2020

Step 1

Step 1: We start by considering the binomial theorem, which states that for any real numbers $a$ and $b$ and any nonnegative integer $n$: $$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$ Show more…

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Prove that for any positive integer $n$ $$\begin{array}{l}\left(\begin{array}{l}n \\0\end{array}\right)-\left(\begin{array}{l}n \\1\end{array}\right)+\left(\begin{array}{l}n \\ 2\end{array}\right)-\left(\begin{array}{l}n \\3\end{array}\right)+\left(\begin{array}{l}n \\4\end{array}\right)-\cdots \\+(-1)^{k}\left(\begin{array}{l}n \\k\end{array}\right)+\cdots+(-1)^{n}\left(\begin{array}{l}n \\n\end{array}\right)=0\end{array}$$

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Key Concepts

Binomial Theorem

The Binomial Theorem explains how to expand expressions of the form (x + y)^n into a sum involving binomial coefficients. It connects algebraic expressions with combinatorial interpretations by showing that each term in the expansion has the form (n choose k) x^(n-k) y^k.

Substitution in Polynomial Expansions

Evaluating a polynomial by substituting specific values into its variables can simplify complex sums. For instance, substituting particular values for x and y in the Binomial Theorem can transform the expansion into an identity, such as showing that the alternating sum of binomial coefficients equals (1 - 1)^n.

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