If x – 1 is an even integer, which of the following numbers must be odd? I. x(x - 1) II. 3(x - 1) III. 2x - 1 IV. 2x + 1 (A) I and II (B) III and IV (C) I and III (D) II and IV (E) I, III, and IV

Q: Let x and y be positive integers such that the prime factorization of x has exactly five 3's and the…

A: x = 35×k          k is an integery = 33×l           l is an integer

Q: For any positive integer n, n(n+1)(2n+ 1) EP = 12 +22 + .. + r?

A:

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

A:

Q: Prove that the statement is true for every positive integer n.

A: Given, 3+9+15+..+(6n-3)=3n2

Q: Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4т + 3 бт + 1

A: Step:-1 Given integer form is 12 k +11 Take  a =12 k +11  then a=12 k +11⇒a=12 k +8+3⇒a=43k +2…

Q: 7.) t + 2(3) + n(n+1) given that any integer for n is positive. ... 3(3) n+1'

A: As per bartleby policy we are supposed to answer only one question per post. This post contains 3…

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: The product of two integers of the form 6k + 5 is of the form a. 6k + 5 b. 6k-3 O c. 7k + 1 O d. 6k…

A: We can give the answer as below

Q: -. For any integer not an integer, then [x]+[m- x] = m-1. m and any real number x, if x is

A: Let m be any integer and x be any real number that is not an integer.

Q: If 5|(2a + 1) for some integer a, then 25|(14a² + 19a – 19).

A:

Q: (3) Let if x 2 Evaluate each of the following, if it exists.

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 6. Let r> 2 be an integer. Show that R(2r, 2r) > 2

A:

Q: Given n < 10" for a fixed positive integer n°2, prove that (n + 1)* < 10*1.

A:

Q: 40. Ifn is any positive integer, then (n- 1)(n-2Xn - 3) ... (3(2X1) = A e B. (n-1y C. n! D. (n-1

A:

Q: P(-1.37<X<1.73)

A: Normal distribution is of utmost importance in statistics . It makes complex problems simpler .

Q: The statement 2n< n! is valid for For only n=4 All positive integers All positive integers n 2 4 All…

A:

Q: Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4m + 3 бт + 1

A: We have to do proper manipulation.

Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

A:

Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: 10. Which of the following expressions cannot equal an odd integer? I. the product of two odd…

A:

Q: If fN is the set of the first 100 positive integers, A is the set of odd numbers between 50 and 90,…

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

Q: Show that2/31 + 2/32 + g+ 2/3n = 1 - 1/3nfor all positive integers n.

A: The given series is the geometric series, hence the formula to find the sum of the series helps to…

Q: Show for all integers n> 1 that 7 I (3ªn+l – 5²n-1 ).

A: a) Prove 734n+1-52n-1 for all integers n≥1 by mathematical induction as follows. Basis step: For…

Q: Prove or disprove if x2-1 is odd, then x is even

A:

Q: п For all integers n > 1, 1.2 2.3 п(п+ 1) п+ 1

A:

Q: A. Prove the following by contrapositive: 1. If x is an even integer, then x² is even. 2. If x is an…

A: # we are entitled to solve one question at a time, please resubmit the other question if you wish to…

Q: Let A= {X: X is an integer and r24}then A= {X: X is an integer and X<4} {X: X e N and X<4} a {X: X…

A: A detailed solution is given below. Z --» the set of integers.

Q: For each positive integer n 2 0, let P(n) be the formula 3n+1 – 1 1+3 + 32 + ... 3n 2

A: We prove this formulas by using PMI

Q: Suppose that f (n) = f (n∕5) + 3n2 when n is a positive integerdivisible by 5, and f (1) = 4. Finda)…

A:

Q: 2. Suppose that f(n) = f(n/3) + 1 when n is a positive integer divisible by 3, and f(1) = 1. Find a)…

A:

Q: True or False: There are integers x, y such that 15 = 215x + 497y. O True O False

A:

Q: Use the factor theorem to verify the statement

A: Consider the statement  x−y is a factor of xn−yn for every positive integer n.

Q: The average pollen count in a certain city on day x of the pollen season is  P(x) = 4x − 0.2x2 for…

A: The average pollen count in a certain city on day x of the pollen season is P(x)=4x-0.2x2…

Q: While proving (a*2(2a+1 is an integer for the case a = 6q +1 which of the following will be obtained…

A: We are given with a(a+1)(2a+1)6 We need to substitute a=6q+1

Q: 6. For an arbitrary integer a, verify the following: (a) 2|a(a + 1), and 3| a(a + 1)(a + 2). (b)…

A:

Q: B={xx is an integer and-1<x<1}

A:

Q: 2. How many positive integer solutions are there to Xị + x2 + x3+ x4 < 100

A:

Q: 1. Prove the following using direct proof. a. If 5x – 3 = x + 5, then x = 2. b. If x and y are both…

A:

Q: 1. If a is an odd integer, then a2 +3a+5 is odd. (use direct proof) 2. Suppose x and y are even. If…

A: Given:1. If a is an odd integer , then a2+3a+5 is odd2. Suppose x and y are even.If x i even, then…

Q: R.4. Prove that for any positive integer n, (6) + 2 (4) + 2² (5) + 2³ (3). ... + 2" = 3".

A:

Q: 7. How many integer solutions are there to 2r, + 2tz + 2rz + Xg +X5 = 9 with x; 2 0? (Hint: break…

A: Solutions to the equation are the set of values for which equations returns the same value on both…

Q: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

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Q: 18. For every positive integer n, 2n < 2".

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Q: 7 and for n > 2, an = an-1 Let, a1 = express an in terms of n.

A: For a geometric progression, the ratio of two consecutive terms is a constant. The nth term, an in a…

Q: Check that o(-2)" (":) = 1– 2(;) + 4(;) + ... + (-2)" (") + ... + (-2)" in ir=0 cannot be greater…

A: Given sum is  ∑r=0n-2rnr=1-2n1+4n2+...+-2rnr+..+-2nWe need to prove that the given sum cannot be…

Q: 3 For all integers n, if n° – n + 1 is even, then n° – n+ 3 is odd. For every nonnegative integer n,…

A:

Q: 4 If x is an integer, then xr >x.

A: See the detailed solution below.

Q: a) Find the number of positive integers not exceeding 120 that are divisible by 3 or 4. 1 0 01 0 0 0…

A: As per our guides lines I answered only 1st part

Q: Given that 3 1 1 2 ]=[3 3 1 -3 2 %3D 1 2 -3 [ 3 -3 find an explicit formula for 2 -4 for any…

A:

Q: Use the factor theorem to verify the statement

A: Given statement is, x+y is a factor of xn-yn for every positive even integer n.   So, Let fx=xn-yn,…

Statements and Logic

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