SOLVE STEP BY STEP IN DIGITAL FORMAT Prove by induction that 1² +2² +3²+...+ n² n(n + 1)(2n + 1) 6

If k is a positive integer, what is the remainder when (k+ 2)(k3 - k) is divided by 6? Saati lo abrirlovd nedt enorm 801 el todmun terW se (A) O (B) 1 (C) 2 (D) 3 (E) 4 ST (AI AA (8) sar (0) ars 10

Prove by induction that (2n+1)3 is an odd number for all positive integers n.

Use mathematical induction to prove that n3 − n is divisible by 3, for every positive integer n.

Show that the product of two numbers of the form 4n +1 is still of that form. Hence show that there are infinitely many primes of the form 4n + 3.

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 9+19+29++ (10n-1) = n(5n+ 4) 2 To prove that 9+19+29++ (10n-1) = n(5n+ 4) is true for all natural numbers n, first show that this statement is true for n= Simplify each side of the statement for the appropriate value of n. 9+19+29++ (10n-1) = n(5n+4) What is the next step in this induction proof? O A. Determine whether the statement holds for any number k + 1. OB. Assume the statement holds for some k, and determine whether it then holds for k + 1. OC. Determine whether the statement holds for some k. If so, it also holds for k + 1.

Prove (n + 6)² – (n + 2)² is always a multiple of 8. (Use direct proof.)

Find the prime factorization of (20) and of 20!.

"For all integers n, n(n? – 1) is divisible by 6." (You may use quotient-remainder theorem and consider cases.) (You should show your work.)

(c) Prove that the product of k consecutive positive integers is divisible by k!