# Mathematical Induction

## Prove that ∑(-1/2)^j = [2^(n+1) + (-1)^n]/3×2^n whenever n is a nonnegative integer

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 […]

## a) Find a formula for 1/(1×2) + 1/(2×3) + 1/n(n+1) by examining the values of this expression for small values of n. b)Prove the formula you conjectured in part (a)

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)

## Prove that 2−2·7+2·7^2 −···+2(−7)^n =(1− (−7)^(n+1))/4 whenever n is a nonnegative integer

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

## Let P(n) be the statement that 1^3 +2^3 +···+n^3 = (n(n + 1)/2)^2 for the positive integer n

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

## a) Find a formula for 1/2 + 1/4 + 1/8 + … + 1/2^n by examining the values of this expression for small values of n. b) Prove the formula you conjectured in part (a).

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

## Prove that 3+3 · 5+3 · 5^2+···+3 · 5^n=3(5^(n+1)−1)/4 whenever n is a nonnegative integer

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 P(n) be the statement that 1^2 +2^2 +···+n^2 = n(n + 1)(2n + 1)/6 for the positive integer n.

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

## There are infinitely many stations on a train route. Suppose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next station. Show that the train stops at all stations

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

## Prove that 1^2−2^2+3^2−···+(−1)^(n−1)n^2=(−1)^(n−1) n(n + 1)/2 whenever n is a positive integer

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

## a) Find a formula for the sum of the first n even positive integers. Prove the formula that you conjectured in part (a)

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