28. Prove that for every integer n 2 1, 1 1+3+5+.·+ (2n-1) 3 (2n+1)(2n+3)++(2n+(2n-1)) 1.

Q: Prove that 1– 2+2² – 2° + -.-+(-1)"2"=2**1 for all nonnegative integers n.

A:

Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

A:

Q: n^2 +1 is not divisible by 3 for every integer n.

A: Any integer n can be written in the form of: n=3a+b, where b = 0 or 1 or 2 Case 1: If b=0, n2+1=…

Q: Prove that 3n+2 and 5n+3 are relatively prime for every integer n.

A: 3n+2 and 5n+3 are relatively prime for every integer n.

Q: Prove that 11! + 22! + ... + nn! = (n + 1)! – 1 whenever n is a positive integer.

A:

Q: Prove that for all positive integers, 1 1 + 3.4 1 ... 1.2 2-3 n(n + 1) n+1'

A:

Q: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

Q: Prove that n! > 2" for n > 4.

A:

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Show that2/31 + 2/32 + g+ 2/3n = 1 - 1/3nfor all positive integers n.

A: The given series is the geometric series, hence the formula to find the sum of the series helps to…

Q: Use induction to prove (2k – 1) = n² – 4 k=3 for all positive integers n2 3.

A:

Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

A:

Q: Prove that 2n+1 +(-1)" 3-2 whenever n is a nonnegative integer.

A:

Q: Prove that the statement is true for every positive integer n.

A: Given arithmetic progression (AP) is: 2+5+8+.........+3n-1 Here the first term is 2 and the common…

Q: 2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

A:

Q: prove that 3^n is greater than n^3 for all n greater than or equal to 4

A:

Q: 4. Prove that if n is an integer and n is a multiple of three (that is, n = 3k for some integer k),…

A:

Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

A:

Q: prove that (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2n for all integers n greater than or…

A: We have to prove that  (1- (1/4))(1-(1/9))(1-(1/16))...(1-(1/ n2)) =(n+1)/2 By mathematical…

Q: Use cases to prove that for all integer n, n2+3n+5 is odd.

A:

Q: Prove that the statement is true for every positive integer n.

A: Given,  2+7+12+...+(5n-3)=12n(5n-1)

Q: Prove that n2 – 7n + 12 is nonnegative whenever n is an integer with n > 3.

A:

Q: Prove that 2n>n² for all integers n25. Prove that 2n<n! for all integers n24.

A: Given below the detailed solution

Q: Prove that for all integers m and n, if m is odd and the remainder is 2 when n is divided by 3, then…

A: Given, m is odd  When n is divided by 3, then the remainder is 2  We have to prove when  3m+2n  is…

Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

A:

Q: snip

A: To Prove n! ≥ 2n for all intergers n≥4We may prove this by Induction Method.

Q: 7. For every integer n z 1, n(5n – 3) 1+6+11+16++(5n– 4) 2

A: Solution is in step 2

Q: Prove that 12 +32 +52 + ...+ (2n + 1)? = (n + 1) (2n + 1)(2n + 3)/3 whenever n is a nonnegative…

A: To prove: That 12+32+52+...+2n+12=n+12n+12n+33 where, n is non negative number. Calculation: Use…

Q: For all integers n>0, prove: 2n < (n+2)!

A:

Q: Suppose that m, n are two integers such that m 2n2 = 1. Does it follow that m and n are relatively…

A: Given that m, n are two integers such that m3-2n2=1. To show m, n are relatively primes we have to…

Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

A:

Q: 2. Show that for positive integers greater than 1, 2 + 6 + 10+...+(4n – 2) = 2n2 -

A: Given query is to find the sum of the sequence.

Q: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

A:

Q: Prove that 3" < n! for all integers n 2 7.

A:

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

A: Sol :- To prove:- 2n+3&lt;=2^n if n is an integer greater than 3 We prove this by induction For n=4…

Q: Prove that n^2+5 >=4n for all integers, n>0.

A:

Q: Prove that the statements here are true for every positive integer n. 1 + 3 + 5 + · · · + ( 2n - 1)…

A: Show that for every positive integer n    1 + 3 + 5 + · · · + ( 2n - 1) = n2

Q: Prove that n² + 1> 2" when n is a positive integer with 1<n<4.

A: Given: is a positive integer with. To prove: .

Q: Prove that 1- 1! +2 · 2! + .. +n ·n! = (n+ 1)! – 1 whenever n is a positive integer.

A:

Q: Prove that n^2 is an even number if and only if 3n + 1 is an odd number by proving  If n^2 is even…

A:

Q: Prove that 3" >n² for all positive integers

A:

Q: prove that n^2 +1 >= 2^n when is a positive integer with 1<=n<=4

A: To prove n^2 +1 &gt;= 2^n when is a positive integer with 1&lt;=n&lt;=4

Q: 6. Prove that for n> 2, 2(;) + (*) =n². п

A: We have too prove the given expression.   According to our guidelines we can solve only one question…

Q: 3n Prove that for every integer n > 0, 9|(4" + 8).

A: To prove: For every integer n≥0, 9|43n+8. Proof by induction method.

Q: 8

A: