use gauss's trick to determine R(n)=2+3+4+...+(n-1)+n+(n+1)

Q: 's Little Theorm and The Pollard Factorization Metod how could I solve Quetion 4 in Section 6.1

A: Introduction: The product of consecutive integers beginning with that natural number and ending with…

Q: 2. (Proof by Contradiction) For every n E Z, 4 t (n² – 3).

A:

Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: 0.3.14: Prove 13 +23 + - +n³ = ()) (mla n(n+1) for all n E N.

A: Mathematical induction

Q: let k >= 2 show that n= 5^k-1 satisfies the equation 4σ(n) = 5n-1

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Q: nl diverges because an > 0, for all n 2 1. ) True False

A: Given statement is

Q: 2. Find the formula of the general n th term . (i) (1,3,5,7,.) (1 2 3 4 (ii) 2 3 45" 11 (iii) 48 16'…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: What is the value of Z if only 1.5​% of all possible Z values are​ larger?   The value of Z if…

A: From the provided information, The value of Z needs to be calculated if 1.5% of all possible Z…

Q: Show that z / [(z-1)(z-3)] = -3 Σ [(z-1)n / 2n+2] - [1 / (2(z-1))] for 0 < |z - 1| < 2.

A: by using techniques of partial fractions and series expansion of 1/(1-x)

Q: Prove that πd/n n = nτ(n)/2.

A: your question has a silly mistake, it should be πd|n d = nτ(n)/2. supposeτ(n)=r , where r be the…

Q: Prove that 12- 22 + 32+ +(-1)"n = (-1)**'n(n+1)/2 for all n E N.

A: Given : Prove that  12−22+32+......+(−1)n+1n2=(−1)n+1n(n+1)2   for n∈ℕ

Q: Prove from the definition: n3 + 65n2 + √n = Θ(n3)

A:

Q: 4. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

A: Given- Let a1 = 1, a2 = 2, b1 = 0, b2 = 1 for n ≥ 3,an = 2an-1 + bn-1for n ≥ 2,bn = bn-1 + an-1 To…

Q: z = ecos(2x)

A:

Q: Use the substitution method to show that the solution of T(n) = 2T(n/3) + n2is O(n2)

A: Given that T(n)=2Tn3+n2 To show that it is O(n2).

Q: Show that for any 1 ≤k≤n. T (1) = n²(n²-1²)...(n² (k − 1)²) 1.3.5 (2k-1)

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Q: B. Using induction, Prove that 2 + 6 + 10 + ... + (4n - 2) = 2n2 for all n e N. %3D

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Q: n-1 n- n-1 n-1 () %3D h - 1 h-p n-1 n-1 P-h = q h-p n-1 ()* (1 -4) n-1 %3D n-1 h - %3D

A:

Q: Prove T(n)= 2T(n-1)+n can be in 0(n2")using represented telescoping method

A:

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: Find the value of K by determines. |1 2 3 0 1 2- |2 1 3 3 4 3 kl

A:

Q: 5. Find and prove a formula for P(n,n-2) for n2 3.

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Q: (5) Show that T(n)o(n) > n for all n E N. 11

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Q: proof It d>4 then ImE IN Such that d-4

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Q: 3. Prove that 6|(9" - 3") for all n E Z+.

A:

Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

Q: Σ (-1)* True/False: We have = -1. (2k)! k=0

A: We know thatcos(x) = ∑k=0∞-1kx2k2k!

Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

A:

Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n n

A:

Q: Employ a suitable c 2n² +3n diverge (a) V5+n² n=1

A: ∑n=1∞2n2+3n5+n5limn→∞an+1an= limn→∞2(n+1)2+3(n+1)5+(n+1)52n2+3n5+n5= limn→∞…

Q: Using induction, prove that 2+ 4 +6+ ... + 2n = n(n+1) for all n E N

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: The principle of mathematical induction can be formally represented as [P(1)V vk( P(k) → P(k + 1))]…

A: Q1.The principle of mathematical induction can be formally expressed as P1∧∀kPk→Pk+1→∀nPn. Hence,…

Q: Prove that: "C. = 2" {1.3.5...(2n - 1)} n!

A: The left-hand side of the expression is: lhs=Cn2n=(2n)!n!(2n-n)!                               Use…

Q: Solve for n if nEN: (4+8+12+ ... +4(n-1)) + (2+4.5+7+. .. 4+5n -%3D126.5 2

A: Sum of n natural numbers is n(n+1)/2.

Q: Prove n 2 2n 1 k = n k - 1 n k=1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: Show that (m) (2n 4n n

A: As n → ∞, By Striling approximation formula n! tends to  sqrt(2 pie n) (n/e)^n

Q: Show that for any n E N, n(n + 1)(2n + 1) %3D k=0

A:

Q: k+1 In k + 2

A:

Q: (In 2x)' In V k.

A:

Q: 3. Suppose a box contains n chips. Chips are successively drawn from the box at random and then…

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Q: Prove by induction:  gn=gn-1+gn-2+gn-4   for n greater than or equal to 4, with g0=1, g1=1, g2=2,…

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Q: Find the possible values of n. (n2 )" – 2n" + 1 = 0 (n +0,1)

A: We have  n2n-2nn+1=0 and n≠0,1 We can write n2n as n2n=n2n=nn2 So, nn2-2nn+1=0 Let nn=x x2-2x+1=0

Q: n! diverges because an > 0. for alln2 1. True False

A:

Q: Prove that E k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A:

Q: 5th term of the G.P 3,6,12 ....is?

A:

Q: If the statement is true, prove. If the statement is false, disprove. 1 + 2" > 3n for all n E Z

A: False.  take n =2

Q: Find all m, n E N such that: n! + 2n + 49 = m²

A: This problem can be solved by graphical approach. Use a graphing utility to plot the graph of…

Q: Use the factor theorem to verify the statement

A: Given statement is, x+y is a factor of xn-yn for every positive even integer n.   So, Let fx=xn-yn,…

Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…