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ID: 3757037 • Letter: P

Question

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6. Prove by induction that the sum of the first n positive even integers is n2+n; Rephrased: Vn P(n) where P(n)-T2i-n2+n i-l

Base Case: For n = 1,

LHS RHS

========= ============

= 2*1 = 2 = 12 + 1 = 2

Clearly, LHS = RHS for n = 1.

Induction Hypothesis: Let us assume that this equality holds true for n, i.e. P(n) = n2 + n. (Condition 1)

Inductive Step: We would now need to show that the equality is true for P(n+1) too, i.e.,

P(n+1) = (n + 1)2 + (n + 1)

Proof:

LHS

=================

= P(n+1)

= Sum(2*i) for i = 1 to n+1

= n2 + n + 2*(n+1) [Using condition 1]

= n*(n + 1) + 2*(n+1)

= (n + 1) * (n + 2)

RHS

================

= (n + 1)2 + (n + 1)

= (n + 1) * (n + 1 + 1)

= (n + 1) * (n + 2)

Clearly, LHS = RHS. Hence, our asusmption is correct. (proved)