3. Prove that 6|(9" - 3") for all n E Z+.

Q: Which of the following is correct? 1 A. X(z) of B. X(z) of -n7"u(-n-1)= C. X(z) of -n7"u(-n − 1) =…

A: Note: Hi! Thank you for your question. As there are multiple questions so as per our guideline we…

Q: Show that ø(n²) = nø(n) for every n E Z*.

A: I have used only one formula of euler phi function

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: For any n e N, let An = {x € R | –n <x < n}. Prove that NneN An = A1. (Recall that we define 0 ¢ N.)

A:

Q: Prove that for all n E N, 1 1x 2 x 3 n(n + 3) 4(n + 1){n + 2 1 1 2 x 3 x 4 n(n + 1)(n+2)

A:

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: 3. Prove the Archimedean Property, if x ER. then there exists K E N such that r SK.

A:

Q: (b) Define fo = 0, f₁ = 1 and fn = fn-1 + fn-2 for n ≥2. Show that 3 fak for all k ≥ 0.

A: As per the guidelines we shall solve one question only as you have asked more than one different…

Q: Prove that 1-21 +2-22 + ……+n· 2" = (n – 1)2"+1+2 for any n E N. -

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Prove by mathematical induction that for n E N, 4r2 - 1 2n + 1 T=1

A:

Q: - Prove that for all n E Z, we have ged(27n – 20, 4n – 3) = 1.

A: Solution

Q: Q3: Prove that (9":n e Z} S {3":n Ee Z} but {9":n e Z} (3":n E Z}.

A: To prove:  {9n:n∈ℤ} ⊆{3n:n∈ℤ} but {9n:n∈ℤ} ≠{3n:n∈ℤ}

Q: C. Prove the following using PMI. 1. 1+4+7+... + (3n-2) = " (3H=1 for all n21. n (n+1) 2. 1+2+3+...…

A: We will prove the given statement using PMI.

Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

A:

Q: (-1)" -Σ S. n=1 Vn+1

A:

Q: - Prove the following identity n n = 0. k=0

A:

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: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

A:

Q: 1. Use PMI to show that for any n E N, 4 divides 5" - 1.

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: that for n > k we Sn+1 ||

A: The expression given is ,  n+1k+1=∑j=knnn-jjk The number of ways to partition n+1 objects into k+1…

Q: Show that for all n E Z, n² 5.

A: Consider the given condition: This shows that whatever will be the values of n, it will always be…

Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

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: Prove n (2n = n n 1 k=1

A:

Q: Prove that ( n2 + 1) / (n + 1) diverges to +∞

A: Let

Q: (5) Show that T(n)o(n) > n for all n E N. 11

A:

Q: Prove the following in Z[i]: if a y and 3|y with a and B relatively prime then aß | y. in

A:

Q: 2. Prove that n²(n + 1)? 13 + 23 +...+ n3 4. for n e N.

A:

Q: 1. Prove that {2" : n E Z} = {,n :n E Z}.

A:

Q: For all n E N prove that 1 3 2n – 1 1 - 2 4 2n V2n + 1

A:

Q: 15. prove that L. d= nd(n)/2

A:

Q: 18. Show that Σkind(k)³ = (Σk|n d(k))² for all positive integer n.

A:

Q: FOR N=195, by fermat's theorem an-2 =

A: Given: FOR N=195, by Fermat's theorem an-2

Q: 2. Show that for every n 2 2 and every 0 <k<n- 2, within the triangle of Binomial Coefficients k+2…

A: We will use properties of binomial coefficient  nCr  +  nCr-1=n+1Cr+1Note : nCr=nr=n!r! n-r!   Proof…

Q: Suppose m=nd where n is odd and 1<n<m. Prove that 2m+1 is composite.

A:

Q: For xn = 5n² + 9n³ + 1, an = en = n², prove or disprove: an = O(an) (n → 0)?

A:

Q: Show that for any n E N, n(n + 1)(2n + 1) %3D k=0

A:

Q: 6. Prove o (k2) = k · Þ(k) for all k E N.

A:

Q: 8. Prove 3n e R|n? – n < 0

A:

Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: Prove E(2k – 1) = n² for n > 1. - k=1

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

Q: Use Euler-Maclaurin's formula to prove that E n² = "(n+ 1)(n +2) 6. %3D 1

A:

Q: (a) Show that for x > 0, 1 5a' < In(1+ x) < |

A: As per guidelines, we will solve only first part. Kindly repost other part separately.

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: 4. Prove that n(3n – 1)r x + 4x + 7x + ... + (3п - 2)x for n e N.

A:

Q: Let f(n) = 2n – n? + 10n – 7. Show that f(n) is O(n³) using specific values of C and ng-

A:

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.