In this case, we will use Mathematical Induction. PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P […]
Prove that ∑(-1/2)^j = [2^(n+1) + (-1)^n]/3×2^n whenever n is a nonnegative integer Read More »
Let’s solve this exercise! a) Find a formula by examining the values of this expression for small values of n n=0 it is undefined. n=1: 1/(1×2) = 1/2 n=2: 1/(1×2) + 1/(2×3) = 1/2 + 1/6 = 4/6=2/3 n=3: 1/2 + 1/6 +1/12 = 3/4 n=4: 3/4 +1/20 = 4/5 1/(1×2) + 1/(2×3) + 1/n(n+1)
To make this proof, we will use the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement
PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k + 1) is true for all positive
We will use the mathematical induction to answer b). So, let’s start with the definition of the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is
PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k + 1) is true for all positive
PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k + 1) is true for all positive
Let’s revisit the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION: “To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: BASIS STEP: We verify that P (1) is true. INDUCTIVE STEP: We show that the conditional statement P (k) → P (k
We should always start with the definitions when solving exercises. So, let’s see the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we complete two steps: The definition above is from the textbook Discrete Mathematics and its
After finding the formula in a), we will need to use induction to prove that it holds for the first n even positive numbers. So, let’s refresh the principle of mathematical induction. PRINCIPLE OF MATHEMATICAL INDUCTION To prove that P(n) is true for all positive integers n, where P (n) is a propositional function, we
