9. Establish the Bernoulli inequality: If 1 + a > for all n ≥ 1. (1+a)" 0, then 1+ na

Q: 1. Express the sum 2.019+2.020 into a fraction of two natural numbers. (2.019 = 2.019191919.. .) 2.…

A: As per the guidelines we can solve 1 problem only.

Q: For all a > 0 and b> 1, the inequalities In (n) ≤n, n < b are true for n sufficiently large (this…

A: The main objective is to write an inequality comparing (ln(n))8 and n. The expression ln n is…

Q: 3. If x >0 show that |(₁ + 2) ¹/³² - ( ₁ + 1 ² - √ ² ) | ≤ 3 ²². )1/3 S Use this inequality to…

A: We recall the binomial theorem for fractional index.

Q: 5. Let r <0 be a negative real number. Use the Archimedean property to prove that there is an n EN…

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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: Compute the value of the following. (Assume n is an integer n (3), for n ≥ 3

A: Given the expression is                     n3, for n≥3. To find the value of the given expression.

Q: For all a > 0 and b> 1, the inequalities In (n) ≤n", n < b are true for n sufficiently large (this…

A: The main objective is to write an inequality comparing (ln(n))8 and n. The expression ln n is…

Q: Find the intersection of  {y|y greater than or equal to 6} n {y|y < 0}

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Q: Solve (r – 2)* (r + 3)° (r - 4)* s 0. Express your answer in interval notation.

A: (x-2)3x+32x-44≤0Value of function must be less than or equal to zero .x+32 &amp; x-44 is always…

Q: Use algebra to find the largest possible value of 8 or small- est possible value of N that makes…

A: 23. This equation comes from the ε,δ definition of the limit. As x→2 the function 3x-1→5. So for all…

Q: 1-Solve the following inequality algebraically. X³-2x²+x-3>2x³+x²-x+1 (.

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Q: 8. Prove the arithmetic mean geometric mean inequality, i.e., a1+a2 + • · · +an > Vaja2 · · · an…

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Q: 3. Prove that 3 € N, \n € N, [n > k] ⇒ [1000m² + 10 ≤ n³]. Hints: you can divide both sides by n²…

A: Answer 3 Given, a statement: It is required to prove the existence of a natural number  such that…

Q: For all a > 0 and b> 1, the inequalities In (n) ≤n, n < b are true for n sufficiently large (this…

A: Please see the attachment

Q: ve that x is odd if and only if 3x +1 is even.

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Q: 1.20 Show that, if 0 < |7₂| < |x₁] then, for all n ≥ 1, |T1| |I₁|-|1₂|| n 12 I in-1

A: Given that Let, Now consider geometric series As

Q: Indicate the smallest integer n such that (6x²+4x)³ f(x) = € 0(x¹). 2A-4x On=4 O n=2 O n=1 O n=6 O…

A: f(x) = (6x2 + 4x)3/(x4 - 4x)         = {2x(3x + 2)}3/{x(x3 - 4)}        = 8x3(3x + 2)3/{x(x3 - 4)}…

Q: (b) Suppose the 4-digit PINS are randomly distributed. How many peo- ple must be in a room such that…

A: 4-digit PINs are randomly distributed amongst k people. 4 digits can have 4! = 24 possible…

Q: 18. Prove: n ΣΚ k = 0 = n(n + 1)(2n + 1) 6

A: The given problem is to prove the given following identity that sum of squares of first n natural…

Q: n | 0 < k <n n k- 1 k k 1 k = 0 or k = n. %3D

A: An equality is given. We need to establish the given equality. The solution is explained below.