Show by induction that for all n natural numbers 0+1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.

Expert Solution

Colby Messinger answered 1 year ago

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for...

Prove the following by induction: 2 + 4 + 6 + …+ 2n = n(n+1) for all integers n Show all work

Show that among all collections with 2n-1 natural numbers in them there are exactly n numbers...

Show that among all collections with 2n-1 natural numbers in them there are exactly n numbers whose sum is divisible by n.

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2)...

Prove these scenarios by mathematical induction: (1) Prove n2 < 2n for all integers n>4 (2) Prove that a finite set with n elements has 2n subsets (3) Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps

Show that the number of labelled simple graphs with n vertices is 2n(n-1)/2. (By induction way!)

Consider the sequence: x0=1/6 and xn+1 = 2xn- 3xn2 | for all natural numbers n. Show:...

Consider the sequence: x0=1/6 and xn+1 = 2xn- 3xn2 | for all natural numbers n. Show: a) xn< 1/3 for all n. b) xn>0 for all n. Hint. Use induction. c) show xn isincreasing. d) calculate the limit.

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a...

2. [6 marks] (Induction) Prove that 21 divides 4n+1 + 5 2n−1 whenever n is a positive integer. HINT: 25 ≡ 4(mod 21)

(1)Prove 6^(2n)-4^(2n) must be a mutiple of 20 (2)Prove 6^(2n)+4^(2n)-2 must be a multiple of 50

Prove by induction: 1 + 1/4 + 1/9 +⋯+ 1/?^2 < 2 − 1/?, for all...

Prove by induction: 1 + 1/4 + 1/9 +⋯+ 1/?^2 < 2 − 1/?, for all integers ?>1

For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n2...

For all n > 2 except n = 6, show how to arrange the numbers 1,2,...,n2 in an n x n array so that each row and column sum to the same constant.

1a. Proof by induction: For every positive integer n, 1•3•5...(2n-1)=(2n)!/(2n•n!). Please explain what the exclamation mark...

1a. Proof by induction: For every positive integer n, 1•3•5...(2n-1)=(2n)!/(2n•n!). Please explain what the exclamation mark means. Thank you for your help! 1b. Proof by induction: For each integer n>=8, there are nonnegative integers a and b such that n=3a+5b

Question