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
a) What is the first?
Inverse Laplace transforms are used to convert the algebraic solution back into the original time domain after the differential equation is solved
1 + 3 + 5 + 7 + . . . + (2n − 1) = n2.
a) To prove that by mathematical induction, what will be the induction
a) assumption?
b) On the basis of this assumption, what must we show?
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.
The statement is true for n = k:
1 + 3 + 5 + 7 + . . . + (2k − 1) = k2.
Problem 3. The sum of the first n odd numbers is equal to the nth square.
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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.
The statement is true for its successor, k + 1:
1 + 3 + 5 + 7 + . . . + (2k − 1) + 2k + 1 = (k + 1)².
c) Show that.