## Levenshtein Distance

Introduction Levenshtein distance between two strings is defined as the minimum number of single-digit edit operations ( Insertion, Deletion, Substitution ) required to convert one string to another. Single Digit Operations Insertion : “Sprk” -> “Spark” Deletion : “Hello” -> “Hllo” Replace : “Carry” -> “Larry” For example, Example 1 s1 = “gone” s2 = …

## 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 …

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

Given n3 – n is divisible by 3 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 n3 – n = 13 – 1 = 0 0 is divisible by 3. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …

## Prove that n^3+2 is not divisible by 8

Given n3+ 2 is not divisible by 8 ∀ n ∈ N Proof For value of n, we have 2 cases CASE 1 : n is odd If n is odd then n3 is also odd. We know ODD + EVEN = ODD. Therefore, n3+2 is odd. 8 cannot divide an odd number. We can …

## Prove by induction n^3+(n+1)^3+(n+2)^3 is divisible by 9

Given n3 + (n + 1)3 + (n + 2)3 is divisible by 9 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 13 + (1 + 1)3 + (1 + 2)3= 13 + 23 + 33 = 1 + 8 + 27 = 36 36 …

## Prove by induction that 6^n-1 is divisible by 5

Given 6n – 1 is divisible by 5 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 6n – 1 = 61 – 1 = 5 5 is divisible by 5. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …

## Prove n^2+2 is not divisible by 4

Given n2 – 2 is not divisible by 4 ∀ n ∈ N Proof For value of n, we have 2 choice CASE 1 : n is even If n is even, then we can write n as n = 2x, x ∈ N n2 – 2 = 4×2 – 2 = 2( 2×2 – …

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

Given 4n – 1 is divisible by 3 ∀ n ∈ N Proof We will prove the given statement by induction STEP 1 n = 1 4n – 1 = 41 – 1 = 3 3 is divisible by 3. Therefore, the statement is true for n = 1 STEP 2 Let the given statement …