Fill in the blanks in the following proof, which shows that the sequence defined by the recurrence relation + 2* for each integer k z 2 - 1 atisfies the following formula. f- 2" +1- 3 for every integer n21 Proof (by mathematical induction): Suppose f,. f. fg. is a sequence that satisfies the recurrence relation f-f + 2" for each integer k z 2, with initial condition f,- 1. Ne need to show that when the sequence f,, f, fy... is defined in this recursive way, all the terms in the sequence also satisfy the exxplicit formula shown above. So let the property P(n) be the equation f- 2" +1 - 3. We will show that P(n) is true for every integer n2 1. Show that P(1) is true: The left-hand side of P(1) is 22 – 3 , which equals 1 . The right-hand side of P(1) is 1 . Since the left-hand and right-hand sides equal each other, P(1) is true. Show that for each integer k 2 1, if P(k) is true, then P(k + 1) is true: gt+1 - 3 , where f,, f, fa... is a sequence defined by the recurrence relation et k be any integer with k z 1, and suppose that P(k) is true. In other words, suppose that f,- f, -f + 2* for each integer k 2 2, with initial condition f, - 1. This is the inductive hypothesis.] Ne must show that P(k + 1) is true. In other words, we must show that f. 2*=2 – 3 Now the left-hand side of P(k + 1) is f* - f, +2* + 1 by definition of fi, fs, f., + 2* +1 by inductive hypothesis V - 3 by the laws of algebra and this is the right-hand side of P(k + 1). Hence the inductive step is complete. Thus, both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]

Discrete mathematics deals with the separated values of data such as integers. It is widely used in the practical field of computer science and mathematics. By knowing discrete mathematics one can improve their reasoning and problem-solving capabilities.…