3 for all n.

Q: 1 + 1.2 1 1 n for all integers n>1 n+1 3. Prove that %3| 2.3 n(n+ 1)

A: To prove the given statement.

Q: 100 How many solutions to the equation E Xx=3 k =1 have the property that each summand X, is a…

A: Number of solutions: The number of non-negative integer solutions to the equation are . Given:

Q: n = n and n = 1. n Show that in

A: A factorial is the product of consecutive integers starting from 1. It is known that nr represents…

Q: n! %3D if n is non-negative integer and proof r-1 r>t

A:

Q: How many combinations are possible with repeats for C(5,2) O 12 O 15 20 O 10

A: In this question, we will use the formula for combination with repetition.

Q: 1. Prove the following equality by induction: 1- 22 + 32 – 4? +. + (2n - 1)? - (2n)² = -n(2n + 1),…

A:

Q: Q.7) In a party 25 people shake hands to each other (only once). How many n(n-1) handshakes will…

A: Given that  Number of people shake hands each other=25 To find  Number of shake hands will occur.

Q: Which of the following statement is always true? (a) p(2n) = p(n) (b) p(n) is always even. (c) Edin…

A:

Q: n (n + () A k=0 | B, = 0.

A:

Q: n! C(n, r) = r!(n – r)! Here we have n = 13 andr = 2. Substitute these values into the formula and…

A: Combination formula.  C(n,r)= n!/r!(n - r)!

Q: B. Find the value of n: (n+3)t 1. (n+1)! = 20

A:

Q: Show that n 1 i2 i=1 <2 - n

A:

Q: Let n = WEE 5 Find n - 1. n – 1 =

A: Given:- n=-2/5 To find:- Value of n-1

Q: Show the solution that a,+1 < a, for each n E N (-1)" п! n=1

A: Find your answer below

Q: Find the smallest positive integer N≥ 2 that gives remainder 1 upon division by each of the integers…

A:

Q: 1(1!) + 2(2!) + ..+ n(n!) = (n + 1)! – 1. %3D 3.

A: (3) The given expression in the LHS can be written in the form of series as, ∑n=1nnn! Now solve the…

Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

Q: n(n + 1)(2n + 1) 6 k=0

A: In this question, we prove the sum of squares of natural numbers.

Q: Choose all those that represent the sum 5+7+9+11+13+...+101. 101 Σκ k = 5 50 E (2K+ 1) j = 2 50 Σ…

A: We have to find the correct representation of the sum 5+7+9+11+13+⋯+101

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

A:

Q: Find all n such that σ(n) = 72.

A:

Q: (":')-(:)·(,".) n +1 n n - Prove that for all integers n > i > 1. i 1

A:

Q: Prove that n Sk k=1 = n(n + 1)(2n + 1) 6 for all n € N.

A: To prove:We will use mathematical induction to prove this.Let

Q: 1. 1.1! +2.2! +3.3! + ... + n.n! = (n + 1)! - 1 for all n ≥ 1

A:

Q: Show that, for n > 1 2n 1.3.5... (2n – 1),2n 2.4.6...2n 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: 3. Prove: For all positive integers, n, n(n + 1)(n + 2) >i(i + 1) = 3

A: Kindly find the attachment

Q: 1 If the N(A) consists of all multiples of x = , then the rank of A 4 is: а. 3 O b. 2 O C. 4 O d. 1…

A: It is clear that from N(A)

Q: Prove that 1-²-2+2-3- + + 1 n(n+1) = n n+1 for all n E N.

A: The given problem is to prove the given following identity,Given 1/1•2 + 1/2•3 + .. + 1/n(n+1) =…