For each integer n 3, let P(n) be the equation: 3+4+5++n= (n2)(n+3) / 2 (a) Is P

Question

For each integer n 3, let P(n) be the equation: 3+4+5++n= (n2)(n+3) / 2

(a) Is P(3) true? Justify your answer.

(b) In the inductive step of a proof that P(n) is true for all integers n 3, we suppose P(k) is true (this is the inductive hypothesis), and then we show that P(k+1) is true. Fill in the blanks below to write what we suppose and what we must show for this particular equation Proof that for all integers k 3, if the equation is true for n = k then it is true for n = k + 1: Let k be any integer that is greater than or equal to 3, and suppose that _____. We must show that _____.

c) Finish the proof started in (b) above.

Explanation / Answer

a) P(3) ,here n = 3,

LHS = 3

RHS = (n2)(n+3) / 2 = (3-2)(3+3)/2 = 6/2 = 3

LHS = RHS

hence P(3) is true

b)if the equation is true for n = k then it is true for n = k + 1: Let k be any integer that is greater than or equal to 3, and suppose that

3+4+5 ... k = (k2)(k+3) / 2

. We must show that 3+4+5 .. k+ (k+1) = (k+1-2)(k+1+3)/2 = (k-1)(k+4)/2

c)now

3+4+5 .. k+ (k+1) =

= (k2)(k+3) / 2 + (k+1)

= ( (k2)(k+3) + 2(k+1)) / 2

= (k^2+k-6)+2k+2)/2

= (k^2+3k-4)/2

= (k-1)(k+4)/2

hence proved

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