#4 Solve using mathmatical induction Please explain. and the inductive step, sta

Question

#4    Solve using mathmatical induction Please explain.

and the inductive step, state the conclusion, namely that is true for all integers n with n b. the mathematical induction proofs in Examples 1-14 to see be helpful to follow these guidelines in the solutions of the induction. The guidelines that we presented can mathematical induction that we introduce in the exercises Explain why these steps show that this formula is true whenever n is a positive integer. Let P(n) be the that 13 + 23 + ... + n3 = (n(n + 0/2)2 for the positive integer n. What is the statement P(1)? Show that P(l) is true, completing the basis step of the proof. What is the inductive hypothesis? What do you need to prove in the inductive step? Complete the inductive step, identifying where you use the inductive hypothesis. Explain why these steps show that this formula is true whenever n is a positive integer.

Explanation / Answer

a)p(1)

so n=1

1^3 = (1(1 + 1)/2)^2

b)

1^3 = (1(1 + 1)/2)^2

1= 1(1)^2
1= 1

so true

c)

hypothesis:assume that the formula holds for n = k.

so

1^3 + 2^3 + ... + k^3 = [k(k + 1)/2]^2

d) We need to provethe hypothesis for n = k + 1

so

1^3 + 2^3 + ... + (k + 1)^3 = [(k + 1)((k + 1)+ 1)/2]^2 )

e)

= [k(k + 1)/2]^2 + (k + 1)^3

= [(k^2)(k + 1)^2]/4 + (k + 1)^3

= [(k^2)(k + 1)^2]/4 + [4(k + 1)^3]/4

= ( [(k + 1)^2]/4 ) ( [(k^2)] + [4(k + 1)] )

= ( [(k + 1)^2]/4 ) (k^2 + 4k + 4)
= ( [(k + 1)^2]/4 ) (k + 2)^2
= [ (k + 1)(k + 2)/2 ]^2
= [ (k + 1)(k + 1 + 1)/2 ]^2

so, 1^3 + 2^3 + ... + (k + 1)^3 = [ (k + 1)(k + 1 + 1)/2 ]^2

f)

1^3 + 2^3 + ... + (k + 1)^3 = [ (k + 1)(k + 1 + 1)/2 ]^2

so this formula holds for n = k + 1

so by the induction principle

1^3 + 2^3 + ... + n^3 = [n(n + 1)/2]^2 is true for all positive integers

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