(d) Still assuming the parallelogram law, show that for any a a € R, ||ax + y||²||ax - y||² = a(||x + y||² − ||x – y||²). (Hint. For a € N, this follows from part (c) by induction. Show it next for reciprocals of natural numbers, a = 1/n, then for (positive) rational numbers. Use continuity to conclude it is true for all a > 0, then deal with a ≤ 0.) Cu

Q: Let a = 3/2 + 34. (a) Compute a³ and write it in the form pa + q where p and q are integers. Use…

A:

Q: x zza) = xuzh

A:

Q: 7. Evaluate using the residue theorem eir dx x2 +1

A:

Q: For geço (Rd), het ₁ (R²). prove loc g*h = C² (R³) and a B (9xh) = (aºg) th for any multi-index B

A:

Q: re a deck of 10 cards numbered from 1 to 10. are grey and some are white. rds numbered 1, 2, 3, 5,…

A: The probability of an event is the number of favorable outcomes divided by the total number of…

Q: 4x 4 lim,-1* - x-1 In x 4 3 5 1 O 2

A:

Q: (a) Explain with cases why, if a is a real number,  then a ≤ |a|. (b) Explain, if, for any real…

A:

Q: (b) Prove that 1(1!) + 2(2!) + 3(3!) + for all positive integers n. ... + n(n!) = (n + 1)! - 1

A:

Q: Let n be an odd integer, a > 0, and b < 0. Prove that x" + ax + b= 0 has a unique root. Hint: If…

A: Let n be an odd integer, a &gt; 0, and b &lt; 0. Prove that xn + ax + b = 0 has a unique root

Q: 1. Use mathematical induction or strong mathematical induction to prove the given statement. 1 (a).…

A:

Q: (2) Translate the following English sentences into symbols. (a) Every real number has a cube root.…

A: Please go through the solution posted below.And If you have any doubts then please ask me in the…

Q: R² = gen gen

A: see below the reason

Q: Prove that for all positive real a we have 22 +5 20x4 2 4/5. 24 x2 +5

A:

Q: Need

A: Given that, a can be written as base 10 x= a0+10a1+ ......+10nar We know that, If a divides b…

Q: 7. Prove: For neN and F, the nth Fibonacci number, =Fn+1 (1) + (7¹) + ("z ²) + (~³²) +--+ (0) = ² 2…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: 2. Prove that võ ¢ Q. You may also use the following facts without proof. (1) Each rational number r…

A:

Q: Prove that if |x-1| < 1, then x² - 2x + 5| < 8.

A:

Q: a + b (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < < b. (b) Prove by contradiction :…

A:

Q: 6. Prove that x is a rational number if and only if 2x – 1 is a rational number.

A:

Q: Use the Geometric Sum identity to prove that, if r 70, then (1 – x)3 1 =1+ (1 - x) + (1 – x)² + |

A: The objectives to use geometry sum identity to prove the following expression.

Q: (5) 1 کی (a) For all n € N and a + 0, show that (b) For a sequence (x) in R, suppose that there…

A: (a) We have to show that,  ∑k=1na-1ak = 1-1an  ∀  n∈ℕ  and  a≠0. (b) It is given that,  xn is a…

Q: If S is a nonempty subset of the negative rational numbers, then: 3x e S(Vy e S(x > y)). O true O…

A:

Q: For each of the statement below, decide whether it is true or false, and prove your claim. (a) For…

A: Sum , difference, product and division of rational numbers is rational

Q: using this xp')-p(W)- lpcad wshow that recursion relation

A: To Determine :-  Using this recursion relation : x pl' x - pl-1' x = l pl x  show that  ∫-11 Pl x Pm…

Q: For any integer n > 1 find, Hint: Substitute x + √x² + 1 = = U 1 So " (x + √x² +1)″

A:

Q: If x ER, x > 0 and r, s are rational numbers, then (i) x > 0; (ii) for x > 1, x" <xs if r < s

A:

Q: If g(x) = 5 – x³, find g'(0) and use it to find an equation of the tangent line to the curve y = 5 –…

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

Q: Show that √3 € Q [√6].

A: Let 3∈Q6 Then there exists rationals a,b such that 3=a+b6 Consider:…

Q: If S is a nonempty subset of the negative rational numbers, then: Jx e S(Vy E S(x > y)). true O…

A: Note: Rational numbers are countably infinite. So there is no largest rational number and no…

Q: For any two rational numbers a,b ∈ Q, we define a new operation a*b=a+b+ab. (a) Is Q closed under…

A: For any two rational numbers a,b ∈ Q  define a new operation a*b=a+b+ab

Q: Let Q denote the set of rational numbers. Show that card(Q)=x0. What is card(R/Q)?

A: Solution:- Step1:-      It is given that ℚ is the set of rational number  then,        we have to…

Q: 7. Suppose a1, a2,...,an are integers, r is a solution to the equation x" + a1x"- + a2x"- +.. + an =…

A: