# Mathematical Induction

## Prove that the 1+3+5+…+(2n-1)=n^2 for every positive integer

Given 1 + 3 + 5 + 7 + ⋯ + (2n-1) = n2 ∀ n ∈ N Proof We will prove the statement by mathematical induction. STEP 1 In this step, we check if the statement is true for n = 1. LHS = 1 + 3 + 5 + 7 + ⋯ + …

## Prove that 4^n-3n-1 is divisible by 9

To Prove 4n – 3n – 1 is divisible by 9 ∀ n ∈ N Proof We will prove the given statement using mathematical induction. Step 1 First, we will check if the given statement is true for n = 1. 4n – 3n – 1 = 41 – 3(1) – 1 = 4 – …

## Prove that 3^n-1 is a multiple of 2

To Prove 3n – 1 is a multiple of 2 ∀ n ∈ N Proof We can prove the given statement using many ways. We will discuss 2 of them. Direct Proof We know the product of two odd numbers is always an odd number. This implies that the term 3n is always odd ∀ n …

## Prove that n^3+5n is divisible by 6

To Prove n3 + 5n is divisible by 6 ∀ n ∈ N Proof We will prove the given statement using the principle of mathematical induction Step 1 We need to check if the statement is true for n = 1. n3 + 5n = 13 + 5(1) = 1 + 5 = 6 Therefore, the statement is …

## Prove by mathematical induction n(n+1)(2n+1) is divisible by 6

Given n(n+1)(2n+1) is divisible by 6 ∀ n ∈ N Proof STEP 1 : We will check if the statement is true for n = 1 n*(n + 1)*(2n + 1) = 1 * (1 + 1) * ( 2*1 + 1 ) = 1 * 2 * 3 = 6 The statement is true …

## Prove that if n is an integer and 3n+2 is even then n is even

Given If 3n + 2 is even then n is even ∀ n ∈ Z Proof By Contradiction We will prove the given statement by contradiction Suppose 3n+2 is even. Assume n is odd. If n is odd, then we can write n as n = 2k + 1, k ∈ Z Substitute n in …

## Prove by induction that 1+3+5+7+⋯+(2n-1)=n^2

Given 1 + 3 + 5 + 7 + ⋯ + (2n-1) = n2 ∀ n ∈ N Proof We will prove the statement using mathematical induction. STEP 1 In this step, we check if the statement is true for n = 1. 1 + 3 + 5 + 7 + ⋯ + (2n-1) = …

## Prove that 2^n + 1 is divisible by 3 for all positive odd integers n

Given 2n + 1 is divisible by 3 ∀ n ∈ O+ Proof We will prove the given statement by induction. STEP 1 n = 1 2n + 1 = 21 + 1 = 3 3 is divisible by 3. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …

## Prove that n^3+2n is divisible by 3

Given n3 + 2n is divisible by 3 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 n3 + 2n = 13 + 2*1 = 3 3 is divisible by 3. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …

## Prove that n^3-7n+3 is divisible by 3

Given n3 – 7n + 3 is divisible by 3 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 n3 – 7n + 3 = 13 – 7*1 + 3 = -3 -3 is divisible by 3. Therefore, the statement is true for n = 1 …