2. 1(2) (2)3 3(4) n(n + 1) (n+ 1) for all natural numbers n.

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

A:

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: 5. Give a general form for T(n) for n > 2

A:

Q: solve 5.3 (using k(pi-k)/2) and setting k =pi/2 or by your own methods if you have easier ones

A: It is known to us that

Q: 13 e, thank you.

A: e)Given that, sum of the first n terms of the given series is

Q: 9.5 The celebrated golden ratio o is the positive real number number satisfying 1/o = $ – 1. Show…

A:

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

A:

Q: Show for all natural numbers n >= 3 that [(1 + (1/n))^n] < n.

A:

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

A:

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

A:

Q: 3.5. Let n E N. Prove that if n + ÷ < 2, then n² + n2 <4.

A:

Q: 2 3 n a. If n E N, then + + + •+ 3! 4! (n+1)! (Hint: 3! (reads 3 factorial) = 3 x 2 x 1. n! = nx…

A: Induction theory: Let the given statement is P(n). Step-1: We need to check P(n) is true for n=1.(or…

Q: 1 Prove that + n is an integer for any n E N. 6

A:

Q: 5. Show that 2 32 3" 31 3" for all positive integers n.

A:

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

A:

Q: Prove that 3^1 + 3^2 + 3^3 + 3^4 + ... + 3^n = (3^(n+1) -3)/(2) for every n ∈ N

A:

Q: सिद्ध है कि n! > 3 ^ n सभी पूर्णांकों के लिए n> = 7।

A: हमें दी गई समस्या को हल करना है:

Q: n Σ k (k!) = (n+ 1)! - 1. %3D k=1

A:

Q: 1 For all integers n21:+ 1.2 2-3 3. n(n +1) n+1

A: Calculate the left hand side of the given equation. 11·2=12 Calculate the right hand side of the…

Q: п For all integers n > 1, 1.2 2.3 п(п+ 1) п+ 1

A:

Q: 1 2. Let e =. Determine the smallest choice of N > 0 such that N. 6 ' n(n + 1)

A: To determine the smallest choice.

Q: If n is an even integer then 7- n3 ) is an odd integer. ) is an even integer. is sometimes even and…

A:

Q: Prove 1 0, Ø ! n n=1

A:

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: 3- (a) Complete k (-1)k 2 k! (k +n+ 1)! 2k+n = -h(x) k=1

A: First I have written the Bessele's function as in an infinite series form then by making 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: 17 Let n = 11 1 Find n + 1 * n + 1

A:

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

A:

Q: 1 If n is a positive integer, then (D-a %3D n!

A:

Q: 3. Prove that when you select n+1 numbers from {1,2,...2n} two of them Must be consecutive (that is,…

A:

Q: 5. Algebraically show that n-1 k- = n · k k-1

A:

Q: For all integers n> 2, 2" < (n+ 1)!.

A: In proof by mathematical induction, we assume that this is true for n=k. Then, we proof that the…

Q: (4) For all natural numbers n, 4^n - 1 is a multiple of 3

A:

Q: 2. Prove that n²>2n for n >2

A: Note that, the given statement P(n): n2&gt;2n for n≥2 has little correction because if n=2  then…

Q: 8. If n+1, 2n-1 and n+5 are in AP. Find the value of n

A:

Q: Q4) Prove for any natural number n 20 Li 1+2+2+. +2" = 2-1 ミ (n+1) .... (11) 1+2+3+ + n = 2

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

Q: 1. Show that b, = (n!)'/n > 20.

A: Consider n!1n=an1n.

Q: For integers n >1, 2 3 1 + 2! n 1 1- 3! 4! (n+ 1)! (n + 1)!

A: solving the problem using induction:

Q: 2 6. The sum of the first n odd natural numbers is equal to n.

A:

Q: k+1 In k + 2

A:

Q: 04 Prove that m (n-k) = (n+1)- ("-m)n 2m n2 m 2

A: Note: Cr+1n+1=Crn+Crn-1+…+Crn-m+Crn-m-1+…+Crr→1 From identity 1, Crn-m=Crn-m-1+Crn-m-2+…+Crr Now,…

Q: ) 1+3+5+ ...+ (2n – 1) = n² for all n > 1.

A:

Q: Find σ(n) if n = 2^k-1 (2^k - 3) where 2^k - 3 is prime

A:

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: η Σ (1) 5 k=1 k odd = 6" - (-4)" 2

A: Here we prove that ∑k=1k oddnnk5k=6n--4n2

Q: Show that for all n € N, one has 1 2√n+1 <√n+1-√n<; 2√ñ

A:

Q: If n is an even integer then 7-n3 is sometimes even and sometimes odd. O is an odd integer is an…

A: We have to state whether the 7-n3 is an even integer, odd integer, or sometimes even and sometimes…

Q: ove that n’ +5n is divisible by 6 for all n E N.

A:

Q: Show that...

A: To show: That n0=nn for n≥0. Concept used: The combination are shown below: Crn=n!r!n-r!…