After having shown the base case is true, I would state the induction hypothesis :
Using the recursive definition, we may state:
We have derived from , thereby completing the proof by induction.
This was a sample problem discussed in class that I already have the solution to, but I am having trouble understanding the highlighted section. Please be very detailed with your explanation. Thank you in advance!
Let f be defined recursively by:
Initial condition: f(0)=1
Recursive step: f(n)=2f(n-1)+1
For the function f above, prove that for all n, f(n)=2^{n+1}-1.
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SOLUTION:
Proof by induction:
1) Basis step:
Prove that f(0)=2^{n+1}-1
f(0)=2^{0+1}-1 is true
because 1=1
2) Inductive step:
Inductive hypothesis: Assume that for some n>=0, f(n)=2^{n+1}-1.
We need to use this hypothesis to prove that f(n+1)=2^{n+2}-1.
Proof: Assume that the inductive hypothesis is true,
(I DO NOT UNDERSTAND HOW THE RIGHT SIDE OF THE BELOW EQUATION (2f(n)+1) WAS FOUND)
then f(n+1)=2f(n)+1 (via Recursive definition of f)
=2(2^{n+1}-1)+1 (via Inductive hypothesis)
=2^{n+2}-2+1
=2^{n+1}-1
# END OF PROOF
After having shown the base case is true, I would state the induction hypothesis :
Using the recursive definition, we may state:
We have derived from , thereby completing the proof by induction.