Exercise 8: Let x ER\Z, and suppose x has a finite number of decimal places. Let n be the numbe of decimal places of x. Since x is not an integer, n > 1. Let a be the digit at the n-th decimal place Show that no matter what the value of a is, 3x also has n decimal places. (Proof by cases)

Q: Prove that (-1+ iE)" + (-1- i5" has either the velue 2n+! or the value -2" f ロ ww. š any integer…

A: The cube root of unity is written as:

Q: Give an inductive proof of the following identity where Fn is the math Fibonacci number F0^2 + F1^2…

A: To prove  f02+f12+f22+...+fn2=fnfn+1          1 where fn is the Fibonacci number

Q: 22] calculate the number of combination and permutation of, 5 distinct objects taken 2 at time?

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Q: Let A = {4n^2 + 1|n ∈ Z}. Show that A is a subset of Z but not equal.

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Q: Find all natural numbers in such that n²<z". Prove your assertion.

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Q: find the amount of odd three-digit numbers that can be created. (

A: here given digits are  [0, 1, 2, 3, 4, 5, 6]. so there are total 7 digits . here use counting…

Q: Kample tetx be Let X€2Z. Prove that for cveri integer n2 , x^is cuen × is aven 2.

A: Yes, for n greater than or equal 2(n is a positive integer) if x is even then x^n will also be even.…

Q: 5. Let r > 1 be an integer such that 2" + 1 is prime. Show that r = 2" for some non-negative integer…

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Q: [a, b] = N=1 (a -,b+-) To prove this, show the following: То Show that [a, b](a – ÷,b +- ) for every…

A: Given that, [a,b]=∩n=1∞a-1n,b+1n This can be written as, ∩n=1∞a-1n,b+1n=[a-1,b] The aim is to show…

Q: Prove C(n,0)=C(n,n)=1 for all whole numbers n

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Q: Show that (n) ‡ 14 for all n ≥ 1.

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Q: 5. Let a be a real number, c an integer, c > 0. Prove that

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Q: Let F(n) denote the Fibonacci numbers. Prove the following property for n ≥ 3 directly from the…

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Q: What is Fn * Fn+1 with F representing Fibonacci numbers

A: The objective is to find Fn×Fn+1 where F represent Fibonacci numbers

Q: x : x is a natural number less than 10​}

A: Write in explicit form

Q: Use generating functions to find the number of ways toselect 14 balls from a jar containing 100 red…

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Q: Provide either a proof or a counterexample for  For all positive real numbers x, x^2 - x ≥ 0

A: According to the given information, it is required to prove the given equality or give a counter…

Q: Solve s(4- dx 3x2

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Q: Show that the nondegenerate interval [2, 5] is uncountable.

A: Here first we have to show that  is not countable.If possible let,  is countable ,so A can be…