If nis a positive integer 1 then prove (D - a) n!

Q: 3. Show by mathematical induction that, for all n2 0, the equality +++.+ = 2 – +2 holds. ... 2Th

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Q: 2. Prove that " The sum of two odd numbers is even: using direct proof.

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Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

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Q: Prove dis prove that the or smallest non- commutative ving is of order 4-

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Q: 2. prove that

A: I solved through matrix form.

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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Q: Prove that (1 + x)n≥ (1 + nx), for all natural number n, where x > – 1.

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Q: 17 Unamughs gniwollet and not yhley to toong ito! Prove that N! > 2N for N a positive integer…

A: Please see the below picture for detailed solution.

Q: Express in the form & j(n/2-j) (2) In(3

A: As per the bartleby guidelines for more than one part only one has to be solved. Please upload the…

Q: 2. Show that In (- - ) = ( + 2nn)i where n = | .... .... ..

A: Consider the following expression: x=ln-12-32iex=-12-32icosx+isinx=-12-32i

Q: R² = gen gen

A: see below the reason

Q: Prove: If x is a positive prime integer, then x+7 is a composite integer. (i.e., not prime)

A: See solution below

Q: 6) Prove that ni 7n² for every integer nz4,whereas. n 7 n3 2 farevery integer n$6.

A: We have to proof by induction process

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n>=1 in question number (7) Sum of squares of first n natural numbers…

Q: 4. With the fact of 1² +2²+...+ n² = n(n+1)(2n + 1)/6, prove that n n n(n −1) ΣΣ(i-j)² 6 i=1 j=1…

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Q: Q prove that for ony natural number n,2+4 +2n=1/n+1) %3D

A: The solution is given below.

Q: IM: Prove that n(n+1)(2n + 1) %3D 6.

A: We prove the given statement by using the sum of first n natural numbers.

Q: How many mombers between 50 and 200 can be expressed as 4k-1, where k is an integer?

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Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: 2. Prove that n + 100 > 3" if n is a positive integer with 1 < n < 3. (Hint: Proof by cases)

A: Concept:

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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Q: integer of the form okts is also but not' conversely

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Q: Prove by induction that E-,(i × 2' +1) = 4(n × 2" – 2" +1). 'i = 1

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Q: Let k be a positive integer. Prove ÷VE. Note: do not use induction. j=1

A: The solution are next step

Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

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Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: let s be positive integer, then (2) + (;) = %3D

A: sC5 = s!/5!*(s-5!) And sC6 = s!/6!*(s-6)!

Q: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

Q: 4. Prove that the cube of any integer can be written as the difference of Notice that squ n³ = (1³ +…

A: To prove that cube of any integer can be written as the difference of two sqaures.we have given hint…

Q: Prove using mathematical induction.  1+3+5+⋯+(2n−1)=n^2, where n^2 means "n squared"

A: Principle of Mathematical Induction: Mathematical Induction is a technique of proving a statement,…

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: ition to prove the n positive number. of Your Your answe that 1/x 00 as

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Q: Prove that 2" > 2n Ior all n E N, n 2 3.

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Q: prove

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Q: Show that if a is a positive integer, then Lo-:(---) n(n +a) n=1

A: Let a is a positive integer, the term a can be written as a = n + a – n

Q: prove that (n(nt)* s an integer V impers

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Q: 4. For natural numbers n and k, prove (C) - ()

A: We have to prove

Q: 4. Prove that 3"< n! if n is an integer greater than 6.

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27

Q: Prove by induction: 1- 3h 27 3* 2

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Q: nt 2 GIVEN: *In ( 1+) hai (-1)" (n +2) 2n+1(n + 1) 4. Compute the exact value of )` n=0

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Q: 4. Prove by mathematical induction that = 2+1 1. 20+2¹+2²+...+2"

A: We have to find solutions

Q: 2. Prove that n² > 2n for n >2

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