4. (P11, Рage 203 (Bernoulli's Inequality) Show that for a natural number n and a nonnegative number b (1+ b)" > 1+ nb. (Hint: In the Binomial Formula, set a = 1.)

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: * For each integer n, let Cn = {n²,n³, nª}. How many elements are in each of C-2, C-1, Co, and C3?…

A: The set always contains the distinct elements. Given Cn = n2 , n3 , n4 Consider C-2 = (-2)2 , (-2)3…

Q: Use mathematical induction to prove that 3n²+1 is divisible by 4 whenever n is an odd number.

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Q: 4. Write as a single power first, if possible, and then evaluate. a) 5³ x 5 b) 4⁹ ÷ 47 c) (2¹)² -5 •…

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Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

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Q: Prove by mathematical induction that for each natural number n 27, n! > 3"

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Instruction: Use extra paper for the computation and fill up all the blank based on the value that…

A: Given:         2a-b25 Using binomial theorem: The expansion is given by the following formula:…

Q: n² + n+ 8 2 He has just shown you a convincing proof that if the equation is true for any integer k…

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Q: which member of binomial expression 10 6, of

A: Binomial expression=2x3+x−110According to Binomial theorem, a+bn=∑k=0nnkan−kbk

Q: Prove by Mathematical Induction that: 192^3+2364+ ...+n[n+1)(n+2) = n(n+1)(n+ 2){n+3)/4

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Q: S: Please answer all the subparts correctly as they are both connect in the same questions in word…

A: We are going to prove that the product of two consecutive integers is even.The product of two…

Q: 'Disprove the statement: For any n E N, if n² +3 is prime, then 2n+7 is prime.

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Q: For a natural number n, we define n! = 1-2-3 (n − 1). n. - For example: 2! = 1-2, 3! = 1-2-3 = 6,…

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Q: Example 5-42. (a) Find the Geometric Mean of 2, 4, 8. 12, 16 and 24. (6) If the observations 2, 4, 8…

A: HEre AS PER POLICY I HAVE CALCULATED FIRST MAIN QUESTION  geometric mean =  a1 *a2*......ann

Q: 4. Knowing that 1 + 3 + 5 + …+ (2n – 1) = n², find 1++3 +5 + .…+ 135 =

A: Solution :   Given that  1 + 3 + 5 + ... + (2n - 1) = n2    -----(1)

Q: 3.2÷.8=

A: In math, long division is a method used for dividing large numbers into groups or parts. It breaks…

Q: 500 E(n? -Ve-n+In n²) A) False B) True 500(501)(1001)* 500(501)] ve + 1000 - In 7 2

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Q: (b) Prove that 27 (2⁹⁹ +1).

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Q: SuPpose that15,000 is invested in a savings account that earns i) Find an expression for the amaunt…

A: $15,000 is invested in a savings account that earns 4% interest, compounded quartely.

Q: 33. Let R be the set of all real numbers, x ER and n eN a n 2 2. If (1+ x) > 0, then which of the…

A: Option (C) II only is correct.

Q: Determine which binomial is a factor of 3+ 22 40a +64 O (2 8) O (x- 2) O (a + 2) O (a + 4)

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Q: 27) (x- 1)(x + 1) s0 A) {1} B) [-1, 1] C) (-1, 1) D) (-, -1]U[1, 0)

A: As per the guidelines when multiple question are posted in single pic  then we are supposed to solve…

Q: Number 4 only. Thanks!

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Q: 3.54. If the arithmetic mean of two unequal positive real number 'a' and 'b', (a > b), be twice as…

A: In question, Given that the arithmetic mean of two unequal real numbers a and b be twice of their…

Q: 3. Let a₁ = 2, a2 = 1, and for each natural number n > 2, let an 2an-1-an-2. Prove that for all…

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Q: Using direct proof method, prove the following. If n is an even integer then n²+ 13 is odd.

A: n is even, so take n=2k and put on the expression.

Q: ress the following using factorials and simplify:

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Q: 3|5 - x| +2< 14

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Q: 4. Prove that (a + b) ≤a+b≤ a + b for any real numbers a and b. (Hint: Use cases when a, b ≥ 0; a, b…

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

Q: expand each expression using the Binomial Theorem

A: Given: 19.(x-2)6

Q: Prove this statement Discrete Math There exists a prime number l so that l^2 − l + 11 is also…

A: We have to prove that, "there exists a prime number I so that I2-I+11 is prime". We know the set of…

Q: Use Pascal's Triangle to expand (5 + 2y)*.

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Q: En-3 and a4 = 22, a5 = 126, and a, = 596. Find a1 , a2, and az

A: Given, an=5an-1-3an-2+4an-3 To find the value of a1,a2 and a3.

Q: Q2 1-The Euler's number (e) is a * rational number a. True b. False

A: Rational numbers are those which can be represented in the from of fraction and those which can not…

Q: Find the integer in the interval (7,10) with the property that if we divide it by 4, the result is…

A: Solve the following

Q: Determine the following. 1. AnB 2. AUB 3. |A|

A: