2n 10. Prove that 31 (5 - 1) for every integer n 2 0.

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: 1. Prove that 7(x – 1) 2.

A: Let, f(x)= ex-7x+7 ,for x≥2. Differentiating both sides with respect to x, f'(x)= ex-7≥0 . This…

Q: Prove 22n! for n=4 Prove R

A: According  to guidelines  we solve one question. We solve question 1 by induction method.

Q: Consider the sum (3)² + (8)² + (13)2 + · . + (15n – 2)2. | m 1 1 m(т + 1)] and k2 = (m(m + 1)(2m +…

A: Consider S=32+82+132+ . . . .+15n-22

Q: 1.) Prove every integer n >= 1, 1+5+9...+(4n-3) = n(2n-1).

A: 1.Apply mathematical induction to prove the statement. Let the given statement be Pn:…

Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

A:

Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

A:

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: Prove, for all positive integers n, 1 3 2n – 1 1 - 2 4 2n (3n

A:

Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

A:

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: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

A:

Q: . Prove that 2 +4+6+•…+ 2n = n(n + 1) for all n > 1.

A: According to the given information it is required to prove

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: Show that Ekin d(k)3 = (Ekn d(k))2 for all positive integer n. %3D

A:

Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

A:

Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: Prove that lor n {k=1 positive unteger every K2K = (n-1)2²+¹+2

A: answer is in next step.

Q: Prove that 3 | 4" – 1 where n E N.

A: To prove 3|4n -1  That is 3 divides 4n -1 for every natural number n.

Q: Prove that 10 134n+1 + 115n+2 – 4 for all n E N.

A:

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

Q: integer of the form okts is also but not' conversely

A:

Q: 6. Prove that 2n +5 and 3n+7 are relatively prime for every integer n.

A: What is Relatively Prime Numbers: Relatively prime or coprime numbers are two such numbers which…

Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

A:

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: Show that 2n+1≤2^n  for all n≥3.

A:

Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

A:

Q: 1. Prove that (fn+1fn+3 – fnfn+4)* = 4 for each positive integer n.

A: We need to prove this, fn+1fn+3-fnfn+42=4

Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: 3. Find integers s and t so that 1 7s+ 11t. Show that they are not unique. %3D

A: Apply the division algorithm, we get 11 = 7*1 + 4 7 = 4*1 + 3 4 = 3*1 + 1 Hence, 1 = 4 - 3*1   = 4 -…

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

A:

Q: Prove that 2+1> (n + 2) · sin(n) for all positive integers n.

A: Concept: Principle of mathematical induction: For each natural number n, mathematical induction is a…

Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

A:

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

A:

Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: 1. Prove that 3 + 3 x 5 + 3 x 52 + 3 x 53 + ..+ 3 x 5° = 3(5n+1 – 1)/4 whenever n is a nonnegative…

A:

Q: 1+ 3k 1- 3k] 1 – 3k 1+3k 2 - prove that AK: 2 1 For A : -1 -

A: Given, A=2-1-12   ...i    ....(i)  We need to show that Ak=121+3k1+3k1+3k1+3k    ...ii     Now, A…

Q: Prove that 74n+3 + 2 is a multiple of 5 for all non-negative integers n.

A:

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

A:

Q: 1. Prove by Induction: 2 + 4 + 6+ .. + 2*n n * (n + 1) %3D for all positive numbers ofn>= 0

A:

Q: 6. Show that (") = 3() + 6n (5) +n³. Using Combinatorial Proof.

A:

Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: For n2 1. prove using mathematical induction: 2" s 2n+1 - 2n-1 - 1

A: We have to solve given problem:

Q: 6. Prove that for n> 2, 2(;) + (*) =n². п

A: We have too prove the given expression.   According to our guidelines we can solve only one question…

Q: 3n Prove that for every integer n > 0, 9|(4" + 8).

A: To prove: For every integer n≥0, 9|43n+8. Proof by induction method.

Q: 2 Lk=0 4. Prove that for all positive integers n, п

A:

Q: In an inductive proof that for every positive integer n, n п(п + 1)(2n + 1) Σ 6. j=1 what must be…

A:

Confidence Intervals

When we calculate intervals in probability and statistics, it can be simply termed as finding out a confidence interval. The confidence interval is usually calculated from the data set which is observed. The value of the parameter that needs to be calculated is present here only.