x4. Prove (1-1) (1-1)… (1-급) n+1 for all n 2 2. 2n 22 32 X5 From the homework and examples vou have formulas for

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: O Prove 1+3+ 9+27+ ·+ 3n-1 = 3"-1 for any positive integer n. %3D

A:

Q: How many solutions are there to the equation X1 + x2 + x3 +x4 + x5 = 21 %3D where r;, i =1, 2, 3, 4,…

A: Depending upon the type of equation, the number of solution will vary. If a system contains linear…

Q: Prove 3 | 17k+1-2*5k+3+11k+3 for k>=0 (nonnegative integer)

A: Here in this problem we have to prove that for k≥0 i.e k is a non negative integer…

Q: coprime

A:

Q: n Prove that + 2 + 22 +...+ 2" +...+ 2n = 3" for all integers n 2 0. * ^) xn-ryr and x=4=1.…

A: We have to prove given identity:

Q: 1- - If イ -1 provethat Kt K_9I

A: We have given that a matrix k of order 3*3.

Q: 2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao.…

A: Answer and explanation is given below..

Q: 5.3.2 (a) Suppose that n > 3. Prove that ("+1)ʻ 5.

A: Given:-             n+1n2 < 2 n+1n2 -22<0n+1n+2n+1n-2<0n+1n+2 =…

Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

A:

Q: 6.60. Below is given a proof of a result. Which result is being proved and which proof technique is…

A:

Q: 2) a) Let x be an even integer. Show that x³ + x + 6 is always even. b) In a), we showed that x³ + x…

A: 2. a) Hint: First suppose that x=2k, k is any integer. Then substitute this value in the given…

Q: 2. Let, a1 = 3 and for n 2 2, an = 2am-+1, express an in terms of n. 3. Let, a1 =3, a2 = 4 and for n…

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

Q: a. Prove that the v63 iş irrational b. For all integers, prove that 4 (n2 + 2) c. Prove or disprove:…

A: Note: As per our company guidelines we are supposed to answer the first three sub-parts of the…

Q: 2n+1+(-1)" Prove that for all integers n 2 0, 3.2"

A:

Q: Prove by using proof by contradiction: If (3n)^2 is even, then n is even.

A: Any odd integer multiply by an odd integer is odd. Any even integer multiply by an even integer is…

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

A: The division algorithm states that for any integer, a, and any positive integer, b, there exists…

Q: a) Verify this equation using mathematical induction that for all n2 1, n(3n – 1) 1 +4+7+ ·.. + (3n…

A:

Q: Find k such that, 2 det 4 li0 -8 -5] -1 -7 = 510 k 0 .

A: Given det2-8-54-1-710k0 =510 we have to find k

Q: 3. Prove that for any positive integer n, n is even if and only if 5n + 2 is even. Indicate which…

A:

Q: Given n is a positive integer, prove that n is even if and only if 3n + 5 is even.  You must use…

A: Topic:- algebra

Q: Prove that 12 - 2²+ 32 - ... .. + (-1)a-1n²=(-1)n-1n(n+1)/2

A: The given problem can be solved using mathematical induction:

Q: Result: There are no integers x and y such that X2 =4y+2. a.) how would you begin a proof by…

A: We have to prove" There are no integers x and y such that    x2= 4y+2 "

Q: 3. Let, a1 = 3, a2 = 4 and for n 2 3, an = 2an-1 + an-2 -5, express a, in terms of n. %3D

A:

Q: 5. Solve the following problems: 5a. By using proof by cases method, prove that "For all odd…

A:

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

A:

Q: Prove that 1 + 4 + 7 + ... + (3n-2) = n(3n-1) / 2

A:

Q: Prove by contradiction method : if x>0  then 2 ≤ x + 1/2

A:

Q: п Find a formula for II (1- ) : and verify. k2 k=2

A: We will expand the terms and check which terms are getting cancel each other

Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

A:

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: Using "Proof by Contradiction" to show that, if 2n³ 5n is odd thenn is an even integer.

A: Now, we have to show that if 2n3-5n is odd then n is an even integer by contradiction. Now, Lets…

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: Prove that 4 + 8 + 12 + .... + 4n = 2n (n + 1 ) Please write it clearly

A: Solution : We have given the expression 4 + 8 + 12 + … + 4n = 2n(n + 1) and We need to prove the…

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: ) ) - 1 3 1 3. (a) Without using a calculator, prove that 2 2n +1 Зп + 1 (b) Prove that for all…

A: 3(b) Prove the identity for n = 2. For…

Q: Select each step to complete a proof by contrapositive of the theorem below. Theorem: For every pair…

A: Here we have to prove the theorem that “For every pair of integers x and y, if 5xy + 8 is even, then…

Q: What is the second step to prove that S, = 2+ 22 + 2' +...+ 2* = 2(2* – 1)? Show that S, is valid…

A:

Q: 2. Let, a, = 3, a2 = 4 and for n 2 3, an= 2an-1+ an-2 + 5, express a, in terms of n.

A:

Q: +34x-71 Ifx is real prove that cannot have vabe between 5 and 9. +2x-7

A: Given  x2+34x−71x2+2x−7

Q: If x^3+1/x^3=18√3 prove that:1/x=√3-√2

A: Given,            x3 + 1x3=183

Q: 7. Suppose n is a natural number that is greater than 100. Alex and Jerry each compute the gcd(4n.…

A: Given n is a natural number greater than 100 Alex and Jerry computed the gcd  (4n,20n^2+15) and got…

Q: 6. If k divides |G| then 3H ≤ G such that |H|=k. *True *False

A:

Q: 7. Suppose a1, a2,...,a, are integers, r is a solution to the equation x" + a1x"-1 + a2x"-2 +... +…

A: for question number 7  we use contradiction to prove the statement.

Q: 2. Prove that for all positive integers n, 1² + 2? + 3? + 2 +n = T(n+1)(2n+1)_ a. Show it is true…

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Q: 2n + 1 6n+3 Show, by using congruence, that 11 divides 3 +2 for n21

A:

Q: 3. A divisibility test for divisibility by 13: а. Prove that 13|(10x + y) if and only if 13|(x +…

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Q: Suppose we prove that 2" > n? for n=1 and 2. Knowing that the inequality is true for some n n > 4…

A: