In this article, we will prove that every group of order 4 is an abelian group. Basic Terminologies Group: A binary operation is a group if It is closed. Associative. Existence of an identity element. Each element has a unique inverse. Abelian Group: A group that is commutative is knows as abelian group, i.e., a*b …
Given is not divisible by 5 Proof We will prove that the given statement is true by contradiction. Let . Assume is divisible by 5. Let, where The above equation is a quadratic equation. The roots of a quadratic equation are given by We know that . If then must also . This means that …
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 by mathematical induction n(n+1)(2n+1) is divisible by 6 Read More »
Given If n is not divisible by 3 then n2-1 is divisible by 3 ∀ n ∈ Z Proof For the value of n, we have 3 choices Case 1 : n % 3 = 1 Let, P = (n2-1) % 3 P = ( ( n2 ) % 3 – 1 % 3 ) …
Prove that if n is not divisible by 3 then n^2-1 is divisible by 3 Read More »
Given 2n2 + n + 1 is not divisible by 3 ∀ n ∈ Z Proof We will prove the given statement by direct method. For each value n, we have three choices CASE 1 : n % 3 = 0 S = 2n2 + n + 1 Taking MOD 3 on both sides S …
Prove that if n is an integer then 2n^2+n+1 is not divisible by 3 Read More »
Given If 3n + 2 is odd then n is odd ∀ n ∈ Z Proof By Contradiction We will prove the given statement by contradiction Suppose 3n+2 is odd. Assume n is even. If n is even, then we can write n as n = 2k, k ∈ Z Substitute n in 3n+2 3(2k) …
Prove that if n is an integer and 3n + 2 is odd then n is odd Read More »
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 that if n is an integer and 3n+2 is even then n is even Read More »
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) = …
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 2^n + 1 is divisible by 3 for all positive odd integers n Read More »
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 …