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?
Dot products are algebraic operations that take two equal-length sequences of numbers and return a single number representing their scalar product value
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.