Prove this statement: we have numbers {a1, a2, a3 .,a2n} which are even and positive, but 0 < {a1, a2, a3.,a2n} < 1 (example:0.2, 0.4, 0.002, 0.00000008) and the sum of these numbers are 1. Now prove there always have the product of two neighbor numbers am * am+1 is less or equal to 1/n2

Q: Q2) Use rounding to 6 digits and compute each of the following: a) P(0.01), where P(x) =…

A: Given, P(x)=(ex-1-x)x2 therefore, P(0.01)=(e0.01-1-0.01)0.012                                =…

Q: Show for all natural numbers n >= 3 that [(1 + (1/n))^n] < n.

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Q: 9(x + 6) - 41 = 75 Prove x - 62/9)

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Q: Prove that for all integers n.4n² +6n + 1 is not divisible by 4. Be sure to explicitly show how each…

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Q: 3. From Fermat's theorem deduce that, for any integer n 20, 13| 1112+6 + 1.

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Q: Prove that 1+r+r^2+...+r^n=(1-r^(n+1))/1-r for all n in N when r doesn't equal 1.

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Q: n>=0

A: According to the question,    False. This statement is not true for all integers n &gt;= 0. A…

Q: Given A= 2n + 1-2, show that for any 4, the number of ways you can have the same value of a given…

A: Given: Objective: Show that for any value of , the number of ways to have the same value for a given…

Q: 4.) T(:) = + 2x-] relate to the baseB, = {H(},and B,,

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Q: Va 占>0 !! に

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Q: How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there…

A: Find change all inequality to grater then equal to 0. Then use the theorem

Q: 31,11 +3153,942m =6454415930m^3 Find m. Show all the steps

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Q: п(п+1)(п+2) E}=1jG+ 1) = а. b. Prove that the sum of the first n Fibonacci numbers with odd indices…

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Q: In 3 - In x - In(x + 5) = 0

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Q: Find a formula for +(−1)n. (2). 1 1 ¹ – ²2 ( ² ) + ² ( 2 ) - -· ··- 1 n+1

A: To Find: A formula for .

Q: Prove that (a) For all integers m and n, n + m and n - m are either both odd or both even.

A: Since you have posted multiple questions, according to our guidelines we can solve only first…

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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Q: Prove that 3! 2(n+3) = = n r=0 (-1) 1 n "Cr r+3 Cr

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Q: Which of the following gives the Binet form of the nth Fibonacci number? O Fn = Fn = Fn = Fn =…

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Q: Show that the weighted arithmetic mean of first 'n' natural numbers whose weight are equal to the…

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Q: One of the striking properties of Fibonacci numbers is that gcd(Fm, Fn) = Fd, (*) where d = gcd (m,…

A: 1. We see,        F9 = 34    [Fibonacci series : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,. . . . . ]…

Q: How many zeroes are at the end of 1000!/(500!500!)?

A: Calculate number of zeroes in 1000! and then subtract it from the number of zeroes in (500!500!).

Q: Prove that for each odd natural number n with n greater than or equal to three,…

A: We have a product line called [(1 + frac12)]. (1 - \frac{1}{3}) (1 + \frac{1}{4}) [ ldots left(1 +…

Q: Prove that 3! 2(n+3) = n r=0 (-1) n "Cr r+3 Cr

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Q: Find a formula for the general term a, {5, 8, 11, 14, 17, .} 5n + 3 On+3 O 3n + 2 O 2n + 3

A: We can answer the question as below

Q: 27. Show that if x and y are positive real numbers that have the same first n digits in their…

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Q: If a and b are integers with gcd(a, b) = d Then gcd (-- d.

A: Given a statement

Q: Prove the following + if a Ez then 4 is

A: Divisibility is the basis of the theory of numbers. A number is referred to be divisible by another…

Q: Give a combinatorial proof for the identity 1+2+3+..+n= (",").

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Q: n —(2) - (-) (1)¹(1)(). (n-1)

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Q: (2) Give a proof by contradiction of theorem “If n^2+1is odd, then n is even”

A: The given problem is to prove by using the contradiction of Theorem of the given statement. Given…

Q: If 3x = 6 mod 9, when -2 <x< 8. Then O x=(-1, 3, 5} x=(-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

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Q: Show that Consider the Fibonacci numbers {F} defined by F₁ = 1, F₂ = 1, and Fn+2= Fn+1 + Fn. - Fn =…

A: Introduction: In mathematics, a recurrence relation is a significant connection that expresses the…

Q: Find A*. 8i 1- i A = 16 3+i 2+i x - (; ; ) ? ? ? A* = ? ? ?

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Q: 1. Prove that for all numbers of k, (5*2^(2k+1) - 1)/3 is an odd number.

A: Let the given statement is        For k = 1 we get this as    is an odd number…

Q: (49-12)3/2 +6 - dt

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Q: Show that:

A: Given, nr+nr+1=n+1r+1

Q: why/where/how does the 1 and the 0 about from at the end of the last few steps?

A: By Leibniz rule for differentiation under integration sign is given by, Here we need to…

Q: If 3x = 6 mod 9, when -2 < x< 8. Then O x={-1, 3, 5} x-1, 2, 5} O x=(-1, 2, 3} O x=(-1, 5, 6}

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Q: ) How many,8-distinct/digit numbers can be formed if the numbers áre amultiple of/5 and contain the…

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Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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