4) if n> 1, then (1+x)" > 1+nx for x E R+ (i.e. x is a non-negative real number). MATH151

Q: If a and x are real numbers with 0 < x <a, then an-x" ≤nan-¹(a-x) for any positive integer n.

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Q: O Prove 1+3+ 9+27+ ·+ 3n-1 = 3"-1 for any positive integer n. %3D

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Q: Is it possible to raise a positive number b to a power and obtain a negative number? Is it possible…

A: The given statement is " The possibility to raise a positive number b to a power and obtain a…

Q: If æ is an irrational number, then it is the case that sup{q E Q : q< x} < x.

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Q: (a) For every integer n, the number n² — n must be even. (b) For all real numbers a, a² < a¹. (c)…

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Q: (n-1)! Use Euler's theorem to prove that if n is an odd integer, then n divides 2 -1.

A: Using Euler theorem to prove it.

Q: Solve for the value of "b" in the equation: * (12) + b 11 Your answer

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Q: Step by step else downvote, with clear handwriting. Prove that for any real number x, there exits a…

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Q: (a) Prove that (-1+iv3)" + (-1– iv3)" has either the value 2"+1 or the value –2" if n is any integer…

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Q: true or False For all number x, then |x| > 0.

A: For all number x then |x|&gt;0

Q: If P= (2, 7) and Q= (2, -3), which could be the coordinates of R if triangle PQR is isosceles?  I…

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Q: Suppose a and b are positive real numbers and n is a natural number with n 2 2. (a) Use the…

A: Let a and b are positive real numbers and n is a natural number with n≥2. Use the difference of…

Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: N How many solutions are there to the equation X₁+X2+X3+X4+X5+X6=41, where each x; is a positive…

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Q: n₁ Evaluate the following formula for p₁ - P₂ = 0, x₁ = 64, x₂ = 11, n₁ =311, n₂ = 296, p₁…

A: Given: P1-P2 = 0, X1 = 64, X2 = 11, n1=311, n2 = 296

Q: Let a and b be non-negative real numbers with b+ 0 and p,q > 1 and+= 1, then a show that ab <+

A: Define a function y : [0,∞)→ℝ by yt=1q+tp-t1p, for every t∈ [0,∞). Now differentiating,…

Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: 3. Show that all integers n ≥ 2 can be expressed as 2x + 3y for some nonnegative integers r and y.

A: Mathematical induction is a proof technique used in mathematics to establish the truth of a…

Q: упе y ne N.

A: Suppose x is a real number such that, x+1x is an integer. We have to show that, xn+1xn is an integer…

Q: x2- 11x+24 For all x 8, = ? ? 8-X 8-X 3-x X - 3 X-8 K X - 11

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Q: ind the interval in the real number line which equals 1 U [4+, 20- n n=1 n

A: Answer given below.

Q: (c) Prove that x - 11 |x| + 11 |x² + 36| 36 for all real r.

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Q: For every integer n > 1, 2+4+6+ + 2n = n² + n.

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Q: Give an example of numbers a and b such that |a + b| &lt; |a|+|b|.

A: To give an example of numbers a and b such that

Q: I also found that the expression was divisible by 13 at n=6 and n=10. Is there anyway you could…

A: Any number n can be written in the form          n = 3m, or 3m+1, or 3m + 2 for some integer m.…

Q: If n is a positive integer, then 2 n - [a] »] + [a + ²] + [a + ² ] -- - + [a + ¹ = ¹] - =] = n n =…

A: Given: nis a positive integer. To Prove: α+α+1n+α+2n . . . +α+n-1n=nα

Q: e3x + 6 enx e(n+1)x = 1+ e® + 2 Sx n! (n+ 1)!

A: The given series is : eex=1+ex+e2x2+e3x6+·····+enxn!+en+1xn+1!   Let fx=ex The above function can be…

Q: Show that if n is an odd integer then []

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Q: Which of the following is true? Select one: a. (n+1)= n(n) for any real number b. г(n) = n(n+1) for…

A: The objective of the question is to determine which of the given mathematical statements is true.

Q: n(n-1) 2 this statement to justify the following. (n+³) = - :)-( (equation 1). 2 (n + 3)(n + 2) 2…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.