Prove by induction that 1*2 + 2^2 + 3*3*2^3 + ... + n*2^n = (n-1)*2^(n+1) + 2 for all n = 1, 2, 3...
Added by Martin C.
Step 1
Proving by induction: Base case: For n=1, we have 1*2 = 2, and (1-1)2^1+1 + 2 = 2, so the equation holds for n=1. Inductive step: Assume the equation holds for some arbitrary k, i.e., k*2^k = (k-1)2^(k+1) + 2. We need to show that it also holds for k+1, i.e., Show more…
Show all steps
Close
Your feedback will help us improve your experience
Madhur L and 57 other Calculus 3 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
Simplify. Use mathematical induction and the property $$ \left(\begin{array}{c} n \\ r-1 \end{array}\right)+\left(\begin{array}{l} n \\ r \end{array}\right)=\left(\begin{array}{c} n+1 \\ r \end{array}\right) $$ to prove the binomial theorem.
Sequences, Series, and Combinatorics
The Binomial Theorem
Use mathematical induction to prove that $$\frac{x^{n}-1}{x-1}=\left(1+x+x^{2}+x^{3}+\cdots+x^{n-1}\right)$$
Additional Topics in Algebra
Mathematical Induction
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD