- #1
Chromium
- 56
- 0
P(n): 1 + 3 + 5 + ... + (2n-1) = n^2
Prove P(1)
P(1) = 1 = 1^2
P(1) is true
(A) P(k): 1 + 3 + 5 +...+ (2k-1) = k^2
(B) P(k + 1): 1 + 3 + 5 +...+ (2k-1) + (2(k+1)-1)
or
(2k-1) + (2k + 1) = (k+1)^2
Assuming A, prove B
(2k-1) = k^2
(2(k+1)-1) = (k+1)(k+1)
(2k + 1) = (k^2 + 2k+ 2)
= (k+1)^2
When in comes to the inductive step, does it "differ" from problem to problem? I can always get to the inductive assumption, but then I'm never sure just how to go about proving it.
Prove P(1)
P(1) = 1 = 1^2
P(1) is true
(A) P(k): 1 + 3 + 5 +...+ (2k-1) = k^2
(B) P(k + 1): 1 + 3 + 5 +...+ (2k-1) + (2(k+1)-1)
or
(2k-1) + (2k + 1) = (k+1)^2
Assuming A, prove B
(2k-1) = k^2
(2(k+1)-1) = (k+1)(k+1)
(2k + 1) = (k^2 + 2k+ 2)
= (k+1)^2
When in comes to the inductive step, does it "differ" from problem to problem? I can always get to the inductive assumption, but then I'm never sure just how to go about proving it.