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

Q: 5) Prove that : tanz 올n(를) i-z

A: tan-1z=i ln[(i+z)/(i-z)]/2.

Q: Let denote the n-th prime number. Show that Pn Pn n=1 diverges.

A: Divergence of the series

Q: Prove that n2 > 2n + 1 for n ≥ 3. Show that the formula is true for n = 3 and then use step 2 of…

A: To show that n2 > 2n + 1 for n ≥ 3 using mathematical induction.

Q: Whenever n is a pOsitiv integer (n > 1). (15 points) %3D

A: Follow the procedure given below.

Q: Prove by Mathematical Induction. n п(п + 1)(2n + 1) k2 6 k=1

A:

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

A:

Q: Prove that the sum of (") where i is from 0 to n is 2".

A: According to the binomial theorem, 1+xn=n0x0+n1x1+n2x2+⋯+nnxn……1

Q: ^n for all n>=6

A:

Q: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

A:

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

A:

Q: For sufficiently lazpe n en ns n Zus (en n)

A: According to this question, we know that

Q: show that for all N in N * Σ (2/(n(n+2)=3/2 - 2N+3/(N+1)(N+2)

A:

Q: Show that n!> n3 if n is large enough.

A:

Q: (b) Show that k-1 (k+1) n n+1 for every n ≥ 1.

A:

Q: Prove the formula for the sum of the first positive integers:

A:

Q: IM- Prove that n(n+ 1)(2n + 1)

A:

Q: Prove that 2n > n2 for n ≥ 5. Show that the formula is true for n = 5 and then use step 2 of…

A: Given n≥5

Q: Prove that the sum of the 1st natural n numbers = n*(n+1)/2

A:

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

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: Prove n (2k – 1) = n² - k=1 for every n EN.

A:

Q: Prove 1 0, Ø ! n n=1

A:

Q: If n>1 prove the sum %3D is not an integer. 1无

A:

Q: 11. Prove by mathematical induction that 1 Σ k (k – 1) = (n + 1)n(n – 1) | 3 1sksn

A: Now for n = 1, Now assume that the given statement is true for n = k, where k is some positive…

Q: Show that 2n+1≤2^n  for all n≥3.

A:

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

A:

Q: 2. What are the two largest positive integers n satisfying n - 1 | n²-A?

A:

Q: 6. Verify Euler's Theorem for n = 16 and a = 5. (7 from 6.5)

A:

Q: Prove by Mathematical Induction. n(n + 1)(2n + 1) k² 6 k=1

A:

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

A:

Q: Prove using mathematical Induction: n 1 2. k=1

A: First we assume that result is true for n=k' then we prove that result is true for n=k'+1 .

Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

A:

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: Let n and k be integers with 1<k<n. Show that n n+1 Zk=1 (k - - n

A: Given Solution:

Q: n(n +1)(2n + 1) 1+4+9+...+n? = %3D 6

A: .

Q: 2. Show that for every n 2 2 and every 0 <k<n- 2, within the triangle of Binomial Coefficients k+2…

A: We will use properties of binomial coefficient  nCr  +  nCr-1=n+1Cr+1Note : nCr=nr=n!r! n-r!   Proof…

Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

A:

Q: Prove using mathematical induction that for every nonnegative integer n. ¡(i + 1) +1 (If n = 0, the…

A: Answer

Q: Let n ∈ Z, Is it true 4 | n4 – n? Prove it or counterexample.

A:

Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: Prove E(2k – 1) = n² for n > 1. - k=1

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

Q: 10. Prove by mathematical induction that Σ k (k + 1) = n (n + 1)(n + 2)/ 3 1sksn

A: In induction method, the given hypothesis or formula P(K) is proved by establishing its validity for…

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: tion 4: Prove the following by using the principle of mathematical induction, n is a [3] positive…

A:

Q: show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

A:

Q: 8. Prove by mathematical induction for n = k+1 1 1 + 5.6 1 1 ...... 4.5 6.7 (п+3)(п+4) 4(n + 4)

A:

Q: for n greater or equal to 1  proove that 2+4+6+...+2n=n^2+n by using mathematical induction

A: