Let S= {n | n=5 k+ 4 for some integer k} Let T= {n|n=5 j– 1 for some integer j}

Q: n Prove that E(2j + 1)³ = (n + 1)²(2n² + 4n + 1) whenever n is a positive integer. j=0

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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: 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).

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

A: Mathematical induction

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

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

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Q: 3. Let n E Z be odd. Show that there exist two consecutive integers such that their sum is equal to…

A: Given n be a odd integer. Which is of the form n= 2k+1, k is an integer.

Q: 5. (a) Verify that t(n) = t(n+ 1) = t(n + 2) = t(n + 3) holds for n = 3655 and 4503. (b) When n =…

A: To find-  Verify that τn = τn+1 = τn+2 = τn+3 holds for n = 3655 and 4503 When n = 14, 206, and…

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: Let function f: N →N be defined on the non-negative integers by if n = 0 f(n) = {f(n – 1)+ 5 (…

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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: (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: 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: Let a, b eR with a < b. Assume that a + <b - for all n e N. Show that n n=1

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Q: 1- Let n and r be positive integers such that r≤n. Then (a) P = nx P-1 (c) P = pn a O b O d n-1 (b)…

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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: Show that for any 1 ≤k≤n. T (1) = n²(n²-1²)...(n² (k − 1)²) 1.3.5 (2k-1)

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

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

A: here is the proof

Q: Ex : Express the Sum o+ 2 +4+6+8 in Sigma nata tion. K = -2 %3D -2

A: First we find nth term of AP then replace k by k+3

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

A: The solution is given as

Q: Prove the following inequality (1+z)" > 1+ nx + n(n 120 for any n2 1. Show that for all positive…

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Q: 8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

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Q: (e) Prove that for any non-negative integer n, i 3 n - 3"+1 – (n+ 1)3" + 1] 4 j=0 (f) Prove that for…

A: (e)…

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

A: By using induction.

Q: 5. (a) Prove that for n any positive integer that 3 (n + j) = 0 j=0 (b) Evaluate the sum Σμα) j=1

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

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Q: 5. Let A = {n E Z; n is odd} and B = {n € Z; 3n² + 7 is even}. Prove that A = B.

A: Given that              A=n∈ℤ ; n is odd   and   B=n∈ℤ; 3n2+7 is even

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

A: Given:

Q: Let n be a positive integer, then n+1≥ for all n.T/F

A: We need to show that n+1≥n for all n where n is a positive integer. Lets proof by contradiction

Q: Let c be a number such that |c| 0. Then use the binomial theorem to show that 1 |c"| < 1+ nd 1 nd…

A: According to the given information, Let c be a number such that:

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

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Q: Let the characteristic of F be n =/0 . Then n is prime.

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Q: 4n +14n+1 Let P(n):9+13+...+(4n+ 5) = 2 Show P(k) P(k +1). Is P(n) true Vintegers n21?

A: ANSWER:

Q: Let n  ∈ Z with n>0. Prove that n | (n+1)n  - 1.

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

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

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Q: Let n be a positive integer and f(n) = 2n10 - n51- 53. Then f(n) is None of these

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Q: 4. Prove that f+ f+..+f = fnfn+1, where n is a positive integer and fn is the nth Fibonacci number.

A:

Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: Assume that p > 1. For all n>1 prove that p – 1 (n + 1)P <n'-P – (n+ 1)'-P.

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Q: Prove that for every n E N is a² | (a + 1)^+1. - 1)n+¹ na — (a + 1), a € N. - -

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Q: Let Hn = the sum from i= 1 to n of 1/ifor any n ≥ 1. Prove or disprove that for any n ≥ 1H2n ≥n/4.

A: So, Hn looks like,             We will use mathematical induction to show that,…

Q: 5. Let A = {n E Z; n is odd} and B = {n € Z; 3n² +7 is even}. Prove that A = B.

A: 5.

Q: B. Consider the function b(n) = 4n+3 for all positive integers n. Compute b(1), b(2), b(3), b(4),…

A: Sequence: An arrangement of the set of numbers or objects following a rule is called a sequence. We…

Q: (b) Z/n = (q) if and only if kq 0 for all 1 < k <n.

A:

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

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Q: 2. Suppose that P(A;) 1 for all i = 1, 2, .... Show that P(N A¡) = 1.

A: Given problem Given 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.