Question-68

Question-68

Let P(n) be the statement that 13  + 23  + …. + n3 = (\frac{n(n+1)}{2})^{2} for the positive integer n.

a)     What is the statement P(1) ?

P(1) is the statement

 

b)    Show that  P (1) is true, completing the basis step of the proof.

The left-hand side of the basis step is ?

The right- hand side of the basis step is ?

 

c)  What is the inductive hypothesis?

The inductive hypothesis is the statement that 13 + 23 + ….+ k3 = ?

 

d) What do you need to prove in the inductive step?

You need to prove that 1 + 2 +….. + k + (k+1)^{3} = ?

 

e) Complete the inductive step, identifying where you use the inductive hypothesis.

 

Solution

a) 1^{3} = [\frac{1.(1+1)}{2}]^{2}.

 

b)

The left-hand side of the basis step is = 1.

The right- hand side of the basis step is = 1.

 

c)(k\frac{(k+1)}{2})^2.

 

d) (k + 1 \frac{(k+2)}{2})^2.

For all k  \geq 1.

 

e) Ans The inductive hypotheis replaces the quantity 13 + 23 + …. + k3  from the left-hand side of part (d) , which shows that [13 + 23 + … + k3 ] + (k + 1)3.

           = (\frac{(k +1) (k + 2)}{2})^{2}

 

 

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