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

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: n3 + 3 n5 n4 - Prove that 14 + 24 + ..n* = +. n for all n > 1. 30 -

A: Consider the provided question,

Q: Prove that (1² × 2!) + (2² × 3!) + …· + (n² × (n + 1)!) – 2 = (n+ 2)! (n – 1) ... For all n E N

A:

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

A:

Q: nl diverges because an > 0, for all n 2 1. ) True False

A: Given statement is

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: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

A:

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

A:

Q: b) Prove the following by mathematical induction: n(n+1)(n+2) 1.2+2.3+3.4+..+n(n+1) : 3 for all…

A:

Q: Suppose n E Z. Prove that if n² + 3 is even, then n is odd.

A:

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

A:

Q: prove that 3^n is greater than n^3 for all n greater than or equal to 4

A:

Q: 1 For all integers n21:+ 1.2 2-3 3. n(n +1) n+1

A: Calculate the left hand side of the given equation. 11·2=12 Calculate the right hand side of the…

Q: (1.3) Show that n - n (n – k) . k + (k + 1) k k 1 - for any two positive integers k and n with k <…

A: Let's find.

Q: Prove that then the sequence {"(1+i) belongs to l1. n3"

A:

Q: (b) Prove the following by mathematical induction: n(n+ 1) (п+2) 1.2+2.3+3.4+….+n(n+1) = 3 for all…

A: By using mathematical induction we solve the result as follows:

Q: 2) Let k be any integer greater than 1. Prove that for all n 20, we have: 1 (k – 1)· k +k' +k +...+…

A: Let k be any integer greater than 1. To prove that for all n ≥ 0 , we have:  (k-1)*[k0 +k1 +k2…

Q: Prove the following formula for all n E N by induction: 1+ 2·2 + ... +n· 2"-l = 1 + (n – 1)2".

A:

Q: 1. Prove that Si-2' = (n – 1) - 2"*+1 + 2 for any positive integer n.

A:

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

A: here is the proof

Q: Prove m+n-1 Σ k k k>0 k

A: We know that  nCr=n!r!(n-r)! When k= 0, the LHS becomes 0 When k=1, the LHS becomes…

Q: *3. Prove that l' + 2 + .. +n =n(n+ 1(2n + 1) for all ne N.

A: By using induction.

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

A:

Q: Prove that for all positive integers k < n, the equality n n n =2n-k k k i=k holds.

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: (3n)! is an integer for all n > 0. (3!)n Prove that

A: We will prove it by Mathematical induction. First We show it is true for initial condition n=0…

Q: Prove that k (x)= - = n (n − 1) for integers n and k with 1 ≤ k ≤ n, using a: a) algebraic proof.

A:

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: Show that φ(n2) = nφ(n) for every n ∈ Z+.

A:

Q: Prove that if h> -1 then 1 +nh ≤ (1 + h)” for all non negative integers n

A:

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

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Q: Q4) Prove for any natural number n 20 Li 1+2+2+. +2" = 2-1 ミ (n+1) .... (11) 1+2+3+ + n = 2

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

Q: Show that Σ dn o(n) n for every positive integer n.

A:

Q: 1. Prove that if n e N, then E-1 k(k + 1) = n(n+1)(n+2)_ k=1 3

A:

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: 6. Prove that for all n E w, the equality 0 n = 0 holds.

A: We will prove by induction. We need to prove for all n∈w, the equality 0·n=0 holds Let n=1, then,…

Q: Show that if M is a number less than 1, then the terms of {n/(n + 1)} eventually exceed M. That is,…

A:

Q: Prove by any means that n!/(n − r)!r! is always an integer for 0 ≤ r ≤ n

A: Since nCr = n!/(n-r)!r! So given expression represents combination and it is read by "n chose r"…

Q: Prove that the following inequality: n-1 Eklg k <n² Ig n 2 k=2

A:

Q: Prove the following: for all integers n > 5, n2 < 2"

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

A:

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: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

A:

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

A:

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 that for any natural n the following equality holds (n – 1)n(n+ 1) 1.2+2.3+...+(n – 1) · n =

A: We have to prove that for any natural n,            1⋅2+2⋅3+......+n−1⋅n=n−1nn+13. We will prove…

Q: 8

A:

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: 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…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.