Backpropagation is an algorithm used to train neural networks by calculating the gradient of the loss function with respect to the weights
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.