Compute the following. Assume the values of the variables are restricted so that the expressions are defined. n! / (n - k + 1)!  (Assuming n - k + 1 >= 0)

Q: An integer n is said to be abundant if o(n)>2n. Every multiple of abundant integer is abundant *…

A: We know,  abundant number is a number that is smaller than the sum of its proper divisors. An…

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: 2. Find an explicit formula for аn 3 За,-1+ 10ар-2 and ao %3D 0, ај 3D 7 An where

A: The solution is given as

Q: 2. (Proof by Contradiction) For every n E Z, 4 t (n² – 3).

A:

Q: Prove n n(n + 1)(2n + 1) Σε ん? 6 k=1 for every n E N.

A: Given ∑k2k=1n=nn+12n+16, n∈N

Q: Give a formula for the nth positive even integer: Here is a theorem: 1+ 2+3 + ... + n п(n+1) n = Use…

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Q: If Sn denotes the sum of the first n terms of an AP prove that S30 = 3 (S20-S10)

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Q: n! diverges because an > 0. for all n > 1. True False

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Q: Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

A:

Q: Give an inductive proof that the Fibonacci numbers Fn and Fn+1 are relatively prime for all n ≥ 0.…

A:

Q: Prove that n/n2 +1 is greater than (n+1)/ (n+1)2 + 1

A: If An and Bn be two term. To show that An is greater than Bn it is sufficient to show that…

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

A:

Q: Solve the following recursion relation T(n) = n* T(n-1) if n>1 = 1.…

A: Explanation of the answer is as follows

Q: Show that the following claim holds when for all n > 1 n n(n + 1)(n +2) (+i) = 3 i=1

A: The aim is to show that ∑i=1n(i2+i)=nn+1n+23, for all n≥1. Consider the LHS, ∑i=1n(i2+i) and rewrite…

Q: Assume that S = Σ(-1)"-¹. 999 9 99 8 O 1000 10 1 (n + 1)5 n=1 n=1 Find the smallest integer N such…

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Q: Compute the following. Assume the values of the variables are restricted so that the expressions are…

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Q: b) Prove that all numbers n > 20 can be represented as n = 4 * i +7*j for some non- negative i and j…

A: Prove that all numbers n≥ 20 can be represented as n = 4*i+7*j for some non- negative i and j by…

Q: Write a short proof of the following and give a counterexample If n is an integer, then 2n + 1 is…

A:

Q: M + N = {m+n : m € M and n € N}. Prove that for all n € N, the number sup(M + N) − n is an upper…

A: Given:   If M and N are both bounded subsets of R. M+N :=m+n : m∈M  and  n∈N

Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: Find an explicit formula for an if ao = 1 and an = (n+1)an-1+3" %3D for n 2 1.

A: Consider the provided question, If a0=1  and  an=n+1an-1+3n for  n≥1Now,…

Q: (3c) Let n E Z. Prove that if n is odd, then 4|(n² – 1)

A: here is the proof

Q: 3(3 + 1) n = 3: Show that 1 + 2 + 3 =

A: To show that 1 + 2 + 3 + ...  + n = n(n + 1)2 is true for the given value of n.   Also, to show…

Q: Remind yourself what () means. For this class, we will use the convention that () = 0 if n < k or if…

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

A:

Q: Prove the following using a direct proof: The sum of the squares of 4 consecutive integers is an…

A: To prove that the sum of the square of 4 consecutive integers is an even integer.

Q: Solve the following recurrence exactly and find an expression for T(n): T(n) = T(n – 1) + 7"+1 + (n…

A:

Q: R.4. Prove that for every integer n, n³ is even if and only if n is even.

A:

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: Prove by the principle of mathematical induction that 3n? + 3n + 1 < 2" for n28 Ensure that you…

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Q: Q3/A/ Prove that (") + („",) = (*). n+: \k+1. n %3| k+1

A: Given:

Q: Is the following statement true or false? For every odd integer n, To answer this question, let n be…

A: Let, n be any odd integer. By definition of odd, n=2k+12, for some integer k So, by solution,…

Q: 28. Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(Vd) where d is not…

A: Given [E:D]=2.

Q: Let m and n be positive integers and let k be the least common multipleof m and n. Show that mZ∩ nZ…

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Q: Prove that (n+1)C n is equal to n+1

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Q: (4b) Let n E Z. Prove that if n is even, then n² + n + 2 is even.

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Q: irline sells n= 125 tickets for a flight that holds only 120 passengers. The robability that a…

A: A random variable X is called to follow the binomial distribution if it takes only non-negative…

Q: 2. For each of the following, use a counterexample to prove the statement is false. * (a) For each…

A:

Q: In the inductive step, we assume that for any k > 10 , P(j) is true for any 8 <js (k-2) and we will…

A: The proof by mathematical induction consists of two steps: The proof consists of two steps: The…

Q: Let n ∈ Z, Is it true 4 | n4 – n? Prove it or counterexample.

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

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Q: Check that o(-2)" (":) = 1– 2(;) + 4(;) + ... + (-2)" (") + ... + (-2)" in ir=0 cannot be greater…

A: Given sum is  ∑r=0n-2rnr=1-2n1+4n2+...+-2rnr+..+-2nWe need to prove that the given sum cannot be…

Q: n! diverges because an > 0. for alln2 1. True False

A:

Q: Prove that E k2k = (n – 1)2"+1 + 2 for all n > 1. i=1

A:

Q: Expand the following recurrence to help you find a closed-form solution, and then use induction to…

A: Introduction: Every recurrence relation with prescribed initial condition has a solution that…

Q: (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < a +b < b. (b) Prove by contradiction : For…

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Q: Find an explicit formula for an if a0= 1 and an= (n + 1)an−1+ 3^n for n ≥ 1

A: Answer

Q: b) Prove that 2 < (1+1/n)" < 3 for all n e N Your answer

A: Solution: Consider the given  2≤1+1nn≤3  for all n∈ℕ Let an = 1+1nn, ∀n∈ℕ Then we have to show that…

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.