Q4 of 66 Page 12

If P(n) is the statement “n2 + n” is even”, and if P(r) is true, then P(r + 1) is true

Given. P(n) = n2 + n is even and P(r) is true.


Prove. P(r + 1) is true

Given P(r) is true that means,


= r2 + r is even


Let Assume r2 + r = 2k - - - - - - (i)


Now, (r + 1)2 + (r + 1)


r2 + 1 + 2r + r + 1


= (r2 + r) + 2r + 2


= 2k + 2r + 2


= 2(k + r + 1)


= 2μ


Therefore, (r + 1)2 + (r + 1) is Even.


Hence, P(r + 1) is true


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