4. Use a combinatorial argument to show that () n 1 -2) n + 22 n 2 3 -

Q: 3. Prove the statement "6 divides n³ - n for all integers n> 0".

A: We need to prove that 6 divides n3-n for all integers n≥0.

Q: 1. Prove that (1) (2) (3) + (2) (3) (4)+...+ n(n+1)(n+2) = "(n+1Xn+2}(n+3)

A: Given: - The statement is, (1)(2)(3)+(2)(3)(4)+......+n(n+1)(n+2)=n(n+1)(n+2)(n+3)4 To prove: - The…

Q: 4.6.2. Prove the following inequalities by induction for all n Ee N. (a) 5" +5 < 5n+1 (b) 1+2+3+…

A: (a) The given statement is: 5n+5&lt;5n+1 We need to prove that the given statement is true for all…

Q: 7 8. The sum of two consecutive even integers is of their product. Find these integers. 24

A: Topic- basic maths

Q: (2) Give a combinatorial proof of the following: G)-00-0000-00' (i) justify your answer with…

A:

Q: Show that 1+2+3+… +n = (";").

A:

Q: ^n for all n>=6

A:

Q: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

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

A:

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

A:

Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: 2n-1 1 1 Σ r(r + 1) 2n r=n ||

A:

Q: Use a combinatorial argument to prove the following equality: (C) · k = =n(") k-

A: A combinatorial argument or proof is an argument that involves counting. Here is the proof.

Q: 2. Use the binomial theorem to show that for all integers n 2 1, 3" = 6) + 2 (4) + 2ª (") + 2" (")

A:

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

A:

Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: The factorial of n is the product of all positive integers less than or equal to n. It is…

A: Note:Hey there! Thank you for the question! We have solved this problem using trial-and-error…

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

A: हमें दी गई समस्या को हल करना है:

Q: 4. What is the equivalent recursive definition for am = 12 + (n- 1)3? ||

A: Given,         The equivalent recursive definition for an=12+n-13.We know that,         The general…

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

A:

Q: 10. If the product of N consecutive posi- tive integers is divided by N, the remainder is always (A)…

A:

Q: 3. Give combinatorial proof of a k (") = n () k-1

A: We know that the combination formula: C(n,r) = n!/(n-r)!r!

Q: If 6-digit positive integer is added to a 6-digit positive integer , the sum could have A) 5 digits…

A: A detailed solution is given below.

Q: Prove that n3 + 2n is divisible by 3 for all n E N.

A: We prove that n3+2n is divisible by 3 for all natural numbers n.

Q: [2 14 If 4 2 -3 the values of the minors of the elements of first row are respectively 1 2

A:

Q: 12 11 10 8. 2 (4, 1) -12-11-10-9-8 -7 -6 -5 -4 -3/2 -1 1 2 3 4 5 7 8 9 10 11 12 -2 8-4). -5 -6 -7 -8…

A: This question is related to circle . We will solve this question by using standard centre- radius…

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: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: (d) Prove that 2" +3" is a multiple of 5 for all odd n E N.

A: Use the mathematical induction to prove the given statement.

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

A:

Q: Exercise 2. Use a combinatorial argument to prove É(:)-("*") n+r+ k=0

A: ∑k=0rn+kk=n+r+1r LHS=n0+n+11+n+22+⋯+n+rr       =n0+n+10+n+11+n+22+⋯+n+rr-n+10       =…

Q: 2. Prove by the Principle of Mathematical Induction that 5-2 + 5-4 + 5-6 +... + 5-2n = 24-'(1 –…

A:

Q: Use combinatorial proof to show that 10 (2n - 5 3 n - 2 n | %3D 10 k 10 – k

A: Given:

Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

A:

Q: Prove or disprove: 2 + 8 + 18 + ... + 2n^2 = n^2 + n

A: Please see the explanation below

Q: 16. Find the positive integer that maximizes the sum of the numbers and its reciprocals?

A: Take integer and add them with his reciprocal Diifferenciate and get

Q: Show that 13 + 23 + · · · + n3 = (n(n + 1)/2)2.

A:

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

A:

Q: I am trying to figure out how to make a know-show table for the question If m is an even integer,…

A: We can make a know-show table for the given condition.If m is an even integer, then the expression…

Q: Show that. 1 (n + 1)(n + 2) n=1

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: nd [Po90].

A: answer is in next step

Q: (b). For all positive integers n 2 2. (") = 2 (5) + n².

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: Question 8: Using combinatorial argument, prove that (:) --C) - »C) * 311 +n* 3

A:

Q: 9. Show that if n is a positive integer then n Σ k² ( x ) = n(n + 1)2²−² k k=0

A: Concept: Mathematical induction: Mathematical induction is a technique for proving that a statement,…

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

A:

Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

A:

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!