(b) 3|2" + (–1)"+1 for all n > 1.

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

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

Q: 5. Give a general form for T(n) for n > 2

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: %3D k (n+1) (k-10 (n+1) k +1 (*) n+1)

A: The correct option is the last option. The detailed solution is as follows below:

Q: 10. Show that for all n E Z, n² – 3n +1 2.

A: Use mathematical induction to prove the given statement. Check whether the result is true for n=1.…

Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

A:

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

A:

Q: 0.3.14: Prove 13 +23 + - +n³ = ()) (mla n(n+1) for all n E N.

A: Mathematical induction

Q: For positive integers m > n , show that -2 + n- 1 -3 + ...+ n + n-1 n - 1 + in

A: The right-side of the given equation can be expressed as…

Q: Prove 0 + 1 + 2 + 3 + ...+ n = (n(n + 1))/2

A:

Q: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

Q: The statement 2n< n! is valid for For only n=4 All positive integers All positive integers n 2 4 All…

A:

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

A: Given statement is

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

A:

Q: 3) Find the non-recursive formula for f (n): f (0) = 6, f (n) = f (n – 1) + 15 for n>1

A: Given f(0) = 6 and f(n) = f(n-1) + 15

Q: 10. Show that È k(C) = n2"-1. Using Combinatorial Proof. k=1

A:

Q: N(x) = x² - 2x 5 N(X)=x²-2x-5 N of N

A:

Q: 3 for all n.

A: We have to find the following conditions according to the given information

Q: Find the term indicated in the expansion (x + 2y)6; third term.

A: Find the term indicated in the expansion (x + 2y)6; third term.

Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

Q: For every positive integer n     12+22+32+...+n2=n(n+1)(2n+1)/6 Let S(n)=12+22+32+...+n2 How do…

A: See the attachment

Q: nd the smallest positive integer N such that n+2 3n-4 -}| < < 0.01 for all n ≥ N.

A: We want to find N such that n+23n-1 - 13 &lt; 0.01    for all   n ≥ N

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: Find all the odd positive integers n which satisfy ó(n) = 20. %3D

A:

Q: Find all the odd positive integers n which satisfy 6(n) = 20.

A:

Q: 1 + 2n < n° for all integers n² 2

A:

Q: Show that yn for all n=1, 2₁- ✓

A:

Q: 3- (a) Complete k (-1)k 2 k! (k +n+ 1)! 2k+n = -h(x) k=1

A: First I have written the Bessele's function as in an infinite series form then by making a…

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: Find the least positive integer n for which 2" + 5" - n is a multiple of 1000.

A: To find the value of n substitute n=1,2,3... multiple of 1000 is a certain number divided by 1000…

Q: 4) Find the Term that free from x of the expansion (x2 +10

A: Use binomial expansion:-

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

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

A:

Q: 5. Algebraically show that n-1 k- = n · k k-1

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: H(n) = 0 if n is  less than or equal to 0  H(n) = 1 if   n=1 or n=2  H(n-1) + H(n-2) - H(n-3)  if…

A: The conditions for H(n) is given as,

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

A:

Q: 1. Show that the binomial coefficient 2"Cn is always an even number for all n > 0.

A: Here, the given binomial coefficient is: Cn2n To prove that it is always an even number for all…

Q: Find the coefficient of x^0 in the expansion ((x^2)+1/x)

A: topic - binomial expansion

Q: (b). For all positive integers n 2 2. (") = 2 (5) + n².

A:

Q: What is the coefficient of x° in the expansion of (x + 1)14 + x² (x – 1)13?

A:

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

A:

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

A:

Q: 3n – 5n is n→ o n° - 2n² +1 lim 3

A: It is given that, limn→∞3n3-5nn3-2n2+1.We have to evaluate it.

Q: Which of the following statements are true (as n gets large or n → 0)?

A:

Q: for n greater or equal to 1  proove that 2+4+6+...+2n=n^2+n by using mathematical induction

A:

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…

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.