2. Prove that n + 100 > 3" if n is a positive integer with 1 < n < 3. (Hint: Proof by cases)

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: 6 First Last Prove that 2 1² +2²+...+n² = n(n+i) (²n+1)/6) for positive integer. n. www

A:

Q: Prove that 6 divides n^3 − n whenever n is a non-negative integer

A:

Q: Whenever n is a pOsitiv integer (n > 1). (15 points) %3D

A: Follow the procedure given below.

Q: Prove for any positive integer n, n=floor(n/2) + ceiling(n/2)

A: Given,          n is a positive integer.     To show-                 n=floor(n/2) + ceiling(n/2)…

Q: Establish formula for the sum of First n odd positive integers and Prove it by using Mathematical…

A: Let 2n-1 represent any odd number , where n is a natural number.We need to find the sum of the…

Q: Prove that 3"< n! whenever n is a positive integer greater than 6.

A: Here we are to prove 3n&lt;n ! whenever n is a positiveinteger greater than 6.To solve this problem…

Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

A:

Q: 62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer,…

A: ∑k=1nk=n(n+1)2 , n is odd integer

Q: 5. Justify by mathematical induction that for every positive integer n, 1(1!) + 2(2!) + 3(3!) + · .…

A:

Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

A:

Q: 2. Prove thatn+ 100 > 3" if n is a positive integer with 1 <n< 3. (Hint: Proof by cases)

A:

Q: Show for all natural numbers n >= 3 that [(1 + (1/n))^n] < n.

A:

Q: Prove by induction, '2 divides n2+n whenever n is a positive intege

A:

Q: Prove that 4 < n! for all n 2 10 by induction

A:

Q: Prove that the sum of the 1st natural n numbers = n*(n+1)/2

A:

Q: 3. Let n be an integer, show that if n is even, then n is also even using indirect proof.

A: Given : Let n be an integer.   To show that if n2 is even, then n is also even.

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: Establish formula for the sum of first n odd positive integers and prove it by using mathematical…

A:

Q: Prove: n! > 2"- for any positive integer n that is greater than 2.

A:

Q: 8. * ) Prove for any odd integer n, we have Proof: Let n be an odd integer. We want to show finish…

A:

Q: 7) Prove that for any integer n, n' -n is even.

A: We need to prove that for any integer n, n3-n is even, Now, an integer can be either even or odd.…

Q: If n>1 prove the sum %3D is not an integer. 1无

A:

Q: 22. What is the least upper bound property for nonemptysubsets of real numbers? Does the least upper…

A: Least upper bound property: Every non empty subset of real numbers which is bounded above has a…

Q: Let n be a positive integer. Prove that :)- (:)- +2. +...+n. = n2"-1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: 2. What are the two largest positive integers n satisfying n - 1 | n²-A?

A:

Q: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

Q: Prove or disprove: let n be an integer. if 3 divides n^2, then 3 divides n.

A: Let us consider some cases for proving that n be an integer. if 3 divides n^2, then 3 divides n.For…

Q: 6. Prove that 3" < n! whenever n is a positive integer greater than 6.

A: We need to prove that ,  3n &lt; n!  for every positive integer n greater than 6. We will use method…

Q: Prove using mathematical Induction: n 1 2. k=1

A: First we assume that result is true for n=k' then we prove that result is true for n=k'+1 .

Q: Prove that the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

Q: Let n and k be integers with 1<k<n. Show that n n+1 Zk=1 (k - - n

A: Given Solution:

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: Prove The Pigeonhole Principle(PHP): Theorem. If k is a positive integer and k + 1 or more objects…

A:

Q: Show by using construction that n^2+5n+6 is not prime for any n is a natural number.

A:

Q: Prove that n2 +1 > 2? when n is a positive integer with 1<n<4.

A:

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: - Prove that if n is an integer, then n = +. (Hint: use the proof by cases with respect to even and…

A:

Q: 21. Prove that if n and k are natural numbers such that k < n, then (*:) - (-) - () n k k

A:

Q: Prove using mathematical induction that for every nonnegative integer n. ¡(i + 1) +1 (If n = 0, the…

A: Answer

Q: 3. Use mathematical induction to prove that the sum of the cube of the first n positive integers…

A:

Q: : Use he comparism test to prove hat n-2 Converyes

A:

Q: Use mathematical induction to prove 8^n − 1 is a multiple of 7 whenever n is a positive integer.

A:

Q: 4. Prove that 3"< n! if n is an integer greater than 6.

A: As per our guideline we are supposed to answer only one asked question.kindly repost other question.…

Q: prove that n^2 +1 >= 2^n when is a positive integer with 1<=n<=4

A: To prove n^2 +1 &gt;= 2^n when is a positive integer with 1&lt;=n&lt;=4

Q: 5. Use ordinary induction to prove that for every positive integer n, n°-n is a multiple of 6. Only…

A: To prove that  For every positive integer n, n^3-n is a multiple of 6 by using induction method.

Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27

Q: Prove or disprove the following: n+1 For any odd integer n, H= ()) 2

A: Let  n=2k+1         -  (1) Where k is an integer. Then n is any odd integer. Now, LHS= n24        -…

Q: 5. Use direct proof technique to prove the following theorem: "If n is an odd integer, then n³ + 1…

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

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.