Question
Use the binomial theorem to prove that$$3^{n}=\sum_{k=0}^{n}\left(\begin{array}{l}n \\k\end{array}\right) 2^{k} .$$Generalize to find the sum$$\sum_{k=0}^{n}\left(\begin{array}{l}n \\k\end{array}\right) r^{k}$$for any real number $r .$
Step 1
Step 1: Recall the binomial theorem, which states that for any non-negative integer n and any real numbers a and b, we have: $$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$ Show more…
Show all steps
Your feedback will help us improve your experience
Prabhat Tyagi and 58 other Calculus 2 / BC 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
Use the Binomial Theorem to show that $$\sum_{k=0}^{n} 2^{k} C(n, k)=3^{n}$$.
Counting Methods and the Pigeonhole Principle
Binomial Coefficients and Combinatorial Identities
Use the binomial theorem to prove that the sum of 2^k * "n choose k" = 3^n.
Use the Binomial Theorem to show that $$0=\sum_{k=0}^{n}(-1)^{k} C(n, k)$$
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD