* 30. Let n and k be integers with 1 ≤ k ≤n. Show that n Σ() ( k=1 k n - 1) = (2n+2)/2 - (2n).

Q: n 2n + 2 /2 n 2n k k=1 \k –1, п+1 n -

A: The objective is to prove that ∑k=1nnknk-1=2n+2n+1/2-2nn This theorem can be proved by induction,…

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: In(x²) = 2 ln(x) for all x True O False

A: ln(x^2) =2ln(x) for all x

Q: To show that the general factorial function: (a)= mn (-). (+¹).(+-¹). here m is a positive integer…

A:

Q: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

A: Consider the provided question,

Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

A:

Q: If A = (; A-: )-..R. e M2x 2(R), find A" for any positive integer n.

A:

Q: 5. Prove that, for n> 1, n

A:

Q: . Show that n! > 3ª for n> 7.

A:

Q: 5. (*) If a and b are nonzero integers such that a | b and b|a, prove that a = ±b.

A: .

Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: 81. Prove: 2+4 +6+ + 2n = n(n + 1). ...

A:

Q: (c) Prove that for n ≥ 1, n [27/7/2≤2- j² 1 n

A:

Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: Show that for all n ≥ 1, the n ₁²>(n+¹) " +1\ in

A: We have to show that for all n⩾1 , the inequality                  ∏j=1njj⩾n+12n(n+1)2Since…

Q: 11. Prove that every positive integer n can be expressed in the form n=4k or n=4k+1 or n=4k+2 or…

A:

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: 10. Let n be an even perfect number. Prove that n=2*(2k+1 – 1), for some integer k.

A:

Q: 4. Find the largest value of n such that 2"|16!.

A:

Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: 2. When the expression (156)(205) (557) is evaluated, it ends with a string of consecutive zero's.…

A:

Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

Q: Write a function for the blue graph of a quadratic function in a form: f(x) = a ( b (x + c))² + d…

A: The given parent graph is shown by red and transformed graph shown by blue in the question.

Q: show that the general factorial function: () (¹) (+).. m (a)m here m is a positive integer and n is…

A: Given Data: To show that the general factorial function  αmn=mmnαmnα+1mn.........α+m-1mn here m is a…

Q: 1. Show for all natural numbers n that (2k – 1) = 1+3+ ...+ (2n – 1) = n². k=1

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Q: 6. For any positive integer n > 1 one has: (2n)! (n!)² n+1 <

A:

Q: For how many integers n with 0 ≤ n ≤ 100 is the quantity integer? n(n-1) (2n-1) 12 equal to an

A:

Q: Compute (a) E-ok · k! = (n+ 1)! – 1. (b) D (") = (). k=0 TIL

A: The problem can be solved using concepts of permutation and combination.

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

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Q: n(n-1) 2 this statement to justify the following. (n+³) = - :)-( (equation 1). 2 (n + 3)(n + 2) 2…

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Q: 5. Solve xdx + ydy = xdy – ydx %3D |

A: Introduction :

Q: Let, a1= 3 and for n ≥ 2, an = 2an-1 + 1, express an in terms of n.

A: