(x+4)² inx-2) 9 3 Express In in terms of In x, In (x+4) and In (x-2). 5, Select one: 2 1 9 In x -In (x+4) - In (x-2) 3 O 9 In x + 2 1 In (x +4) – –In (x- 2) 2 9 Inx + 슬in (x+4) --in (x-2) 3 1 9 In x - 2 In (x+4) –In (x-2) Solve the equation log, (2x+3)=1 + log, (x-5).

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: 16. The expansion of (2 + 3x;-1 is valid when >x>

A:

Q: u will prove that 7*+1 - 1 is divisible by 6. ypothesis, since 6 evenly divides , there exists an…

A: We will fill in the blanks.

Q: 26. In how many ways can up to 4 students be selected from 6 girls and 5 boys if each selection must…

A:

Q: 1 Prove that 1+2+3+..+n = +1) hy proof of mathematical induction.

A:

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: 2. Let n be a positive integer. Show that 4n 4n-k (4n 3 4n k=2n (Hint: an argument similar to the…

A: Solve the question by first opening the summation on the left side of the inequality. Simplify the…

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A:

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

A:

Q: Let p be a prime divisor of the Fermat number Fn= 22^n+ 1 i. Show that ordp2 = 2n+ 1 ii. Use…

A: Given p be a prime divisor of the number Fn=22n+1. (i) Since p divides Fn=22n+1, therefore: Fn ≅0…

Q: 1 Prove 2.5 1 1 + .…+ 8.11 1 for n > 1. %3D 5.8 (3n-1)(3n+2)

A: Prove S(n) for n=1.   LHS =1/2.5 =1/10; RHS = 1/6+4=1/10   So it is true for n=1.

Q: 8"-3 is a multiple of 5. In other words, prove that 8"-3"=5 for some i integer j.

A:

Q: Express in the form & j(n/2-j) (2) In(3

A: As per the bartleby guidelines for more than one part only one has to be solved. Please upload the…

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

A:

Q: 1. Prove by induction: 22 + 1*3 12 n2 а. п(п+1) + (2п-1)+(2n+1) ... 3*5 2(2n+1)

A:

Q: Show that z / [(z-1)(z-3)] = -3 Σ [(z-1)n / 2n+2] - [1 / (2(z-1))] for 0 < |z - 1| < 2.

A: by using techniques of partial fractions and series expansion of 1/(1-x)

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: Use a combinatorial argument to prove that EC(n, k) = 2". k=0

A: The Multiplication Principle If one event can occur in m ways and a second event can occur in n ways…

Q: 7. For n > 1, verify that 1? + 3? + 5² + ...+ (2n – 1)? = ( 2" *1 ) %3D -

A: We have to prove that for n&gt;=1 in question number (7) Sum of squares of first n natural numbers…

Q: Determine whether 3 and 5 are irreducible or not in Z[i]. 3 and 5 are reducible. O 3 and 5 are…

A: We have to check

Q: 7) Use a direct proof to show that: a) The sum of an even integer and an odd integer is odd. b) If n…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: 4. Let, a, = 1, az = 2, b, = 0, b2 = 1, for n 2 3, an = 2an-1+ bn-1 for n 22 bn = bn-1+ an-1 express…

A:

Q: Prove that for every non-negative interger n: 2n+1 + (-1)" Σ 3· 2n i=0

A:

Q: 2) Prove that vnEN 12n+1 is in lowest term, i.e., prove that 30n+1 gcd(12n + 1,30n + 1) = 1 3) Vns A

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Suppose n₁, n₂ are positive integers and gcd(01, n₂) = 1. Then for every integer x, if n₁x and n₂|x,…

A:

Q: Show that 1+3/ -5 is irreducible which is not prime in ZlV-5]

A: We know that ,  ℤ-5 = a + b-5 / a , b ∈ ℤ We need to show that , 1 + 3-5  is irreducible but not…

Q: 3. By using mathematical induction proves 2' + 22 + 2 +.. +2" = 2"*! – 2 is true for all positive 3…

A:

Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

A:

Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

A:

Q: 7. Prove that 2. O = (-1). %3D

A:

Q: Determine whether 5 and 7 are irreducible or not in Zi]. O 5 is reducible but 7 is irreducible. O 5…

A: We have to check

Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

A:

Q: о Prove 2 (2n – 1) = n n - п - 1 k=1

A:

Q: Prove that 11 is the only prime number of the form 4n2 - 25.

A:

Q: Determine whether 5 and 7 are irreducible or not in Zi. O 5 is reducible but 7 is irreducible. O 5…

A: Here , both 5 and 7 are irreducible because both cannot be reduced further. As both are neither in…

Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: 1. Prove that 3 + 3 x 5 + 3 x 52 + 3 x 53 + ..+ 3 x 5° = 3(5n+1 – 1)/4 whenever n is a nonnegative…

A:

Q: 1+ 3k 1- 3k] 1 – 3k 1+3k 2 - prove that AK: 2 1 For A : -1 -

A: Given, A=2-1-12   ...i    ....(i)  We need to show that Ak=121+3k1+3k1+3k1+3k    ...ii     Now, A…

Q: part ii and iii, thank you

A: According to pascal law,

Q: first prove that ()" <n! < ()" forn2 6.

A: We will prove                        n3n≤n!≤n3n for n≥6.

Q: Prove that 2n + 3 ≤ 2n if n is an integer greater than 3.

A: Sol :- To prove:- 2n+3&lt;=2^n if n is an integer greater than 3 We prove this by induction For n=4…

Q: Let's take an integer n to be odd if there is some integer k such that n = 2k – 1. Prove that the…

A:

Q: 1. Prove by Induction: 2 + 4 + 6+ .. + 2*n n * (n + 1) %3D for all positive numbers ofn>= 0

A:

Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: 1. Show that 2n n=0 onverges absolutely.

A: Given series: According to the ratio test:

Q: 6. By Principle of Mathematical Induction prove 6n 6

A:

Q: show that the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

A:

Q: 15 of 20 -722 –8x =? limz→00 3–72²+6x+6 O Does not exist O 00 O 7

A:

Q: 1. Prove that 1 + Зп+2 -) = 2 an4) = (;+-) + (;+}-)+- = In3. ... = In3. - Зп+1 Зп+3,

A:

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.