- #1

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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.