Verification and validation are processes used to ensure that a mathematical model accurately represents the system it is intended to simulate and provides reliable results
Problem 1. According to the principle of mathematical induction, to prove a statement that is asserted about every natural number n, there are two things to prove.
a) What is the first?
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1 + 3 + 5 + 7 + . . . + (2n − 1) = n2.
a) To prove that by mathematical induction, what will be the induction
a) assumption?
The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k2.
b) On the basis of this assumption, what must we show?
The statement is true for its successor, k + 1:
1 + 3 + 5 + 7 + . . . + (2k − 1) + 2k + 1 = (k + 1)².
c) Show that.
If the statement is true for n = k, then it will be true for its successor, k + 1.
b) What is the second?
The statement is true for n = 1.
c) Part a) contains the induction assumption. What is it?
The statement is true for n = k.
Problem 2. Let S(n) = 2n − 1. Evaluate
a) S(k)
= 2k − 1
b) S(k + 1)
= 2(k + 1) − 1 = 2k + 2 − 1 = 2k + 1
Problem 3. The sum of the first n odd numbers is equal to the nth square.