A student is trying to prove a statement 𝑄(𝑛)𝑄(𝑛) for all integers 𝑛≥2𝑛≥2 by strong induction. As an induction step they have proved that, for all integers 𝑘≥3𝑘≥3, if 𝑄(𝑘−1)𝑄(𝑘−1) then 𝑄(𝑘+1)𝑄(𝑘+1). What is the minimum number of base steps they will need to finish their proof?

Assume that N = {0, 1, 2, 3, . . .}.(a) Suppose that we change the induction mechanism as follows:• Base case: Prove that P (0) is true• Inductive step: Prove that for all k ≥ 0, P (k) ⇒ P (k + 2)Explain why this would not constitute a valid proof that P (n) is true for alln ∈ N. How would you change the base case to obtain a valid proof?

Using mathematical induction, prove that the sum of the first 𝒏 positive integers isequal to 𝒏(𝒏+𝟏)𝟐 .

16) What is proven in the inductive step of mathematical induction?a) The statement holds for all real numbers.b) The statement holds for an arbitrary case.c) The statement holds for the next case, given it holds for a particular case.d) The statement holds for the base case.

lim𝑥→0(𝑥+1)5−1𝑥

Using mathematical induction, prove that the sum of the first 𝒏 odd positive integersis equal to 𝒏𝟐