2 and an+1 Зап + 2. Find an explicit Suppose that a1, a2, a3, . .. is defined by a1 formula for a, in simplest form

Q: Prove that (-1ti3)" + (-1-is3)" has either the value 2n+' or the value -2" if r is any ). integer…

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Q: Find an explicit formula for an if ao = 1 and an (n + 1)an-1+3" %3D for n > 1.

A: To Determine :-  An explicit formula for an if a0 = 1 and an =  n + 1an-1 + 3n for n ≥ 1.

Q: Give a formula for the nth positive even integer: Here is a theorem: 1+ 2+3 + ... + n п(n+1) n = Use…

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Q: 1. Let, a, = 3, a2 = 4 and for n 2 3, a, = 2an-1+ an-2, express a, in terms of n.

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Q: Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n > 3, an = 2an-1+ bn-1 for n > 2 bn = bn-1+ ɑn-1 express…

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

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Q: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

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Q: Determine whether the statement is true or false. Prove the statement directly from the definitions…

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Q: Find a closed formula for a, if ak = 5n² + 4n k=1 an =

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Q: Select all the expressions that are greater than 3. Be sure to show your work 1.67 + 1.4 2.97 +…

A: We have to select all the expressions that are greater than 3. (1) 1.67 + 1.4  = 3.07 > 3

Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

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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: so-Z of what is 21

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Q: Prove that if for some integers a, b, c we have that a° + 6° + c° is divisible by 9, thên at least…

A: Here we have to prove that if for some integers a,b,c we have that a3+b3+c3 is divisible by 9,then…

Q: Suppose that the product of two natural numbers is odd. A whole number a is even if and only if…

A: Algebra Question

Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

A: We have to factories the equation  x3+3x2+2x-258,474,216=0

Q: Find, If possible, A + B, A- B, 2A, 2A - B, and B +A. (If not possible, enter IMPOSSIBLE In any…

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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…

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Q: a) Verify this equation using mathematical induction that for all n2 1, n(3n – 1) 1 +4+7+ ·.. + (3n…

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Q: EIf a,,a,,a,.--,a, be an A P of non - zero terms, prove that ------ 1 1 +. 1 n-1 %3D a,a, a,a, a,a,

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Q: For each positive integer n, find with proof GCD(220° - 220, 220" + 218).

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Q: 1. Let, a, = 3, a2 = 4 and for n 2 3, a,= 2an-1+ an-2, express a, in terms of n.

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Q: 4. Let, a1 = 3, a2 = 4 and for n ≥ 3, an = 2an-1 + an-2 + n2 , express an in terms of n.

A: As per guidelines we should able to do only one question at a time because both questions are…

Q: Suppose n is a natural number greater than 100. Alex and Jerry each compute the gcd(4n, 20n2 + 15)…

A: Given that n is a natural number greater than 100. Alex and Jerry computed the gcd4n,20n2+15 and got…

Q: Consider the following statement. For every integer m, 7m + 4 is not divisible by 7. Construct a…

A: Given statement is for every integer m,7m+4 is not divisible by 7. We prove this by contradiction.

Q: If a1, az, az,.-ans. be the A.P. and S, = a-1m+1+a-1m+2+a-1n+3 +.--+ am %3D Prove that :…

A: In this question we have to proof S1, S2, S3,.... are in Arithmetic Progression.

Q: . Let, a¡ = 1, a2 = 2, b, = 0, b2= 1, %3D for n 2 3, an = 2an-1+ bn-1 for n 22 b,= bn-1+ an-1…

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Q: To show that r - 1= In(r-r+ 1) has at least two different positive root in R. 6.

A: We need to analyze the roots of the equation x-1=lnx2-x+1. This equation can be re-written as…

Q: Given that 3 1 3 -3 1 1 -4 1 2 -3 | n 3 -3 find an explicit formula for for any non-negative integer…

A: See solution below

Q: Find an explicit formula for an if a0= 1, a1= 2, and an= 2an−1−an−2for n ≥ 2.

A: given a0= 1, a1= 2  and an= 2an-1−an-2for n ≥ 2.

Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

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Q: If the sum of two counting numbers is an even number, then the product of the two counting numbers…

A: ANSWER:- 1) From the given statement, If the sum of two counting numbers is an even counting number,…

Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

Q: If a =(2,3,-1) ,(1,6,0) ,(2,1,3) prove that Aa^-1 =A^-1A=I Please show all the step

A: Given matrix is, A=23-1160213

Q: is a1 = 2, a2 =3, and an  = 5an–1 – 4an–2 for n > 2 expressed in explicit formula? true or false?

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Q: Let, a1 = 1, a2 = 2, b1 = 0, b2 = 1, for n ≥ 3, an = 2an-1+ bn-1 for n ≥ 2 bn = bn-1 + an-1…

A: The Fibonacci sequence can be defined as a series where a term of the sequence is the sum of the…

Q: A sequence {am} of real numbers is defined recursively by a1 = 1, a2 = 2 and a, = (i–1)a; for n > 3.…

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Q: Consider the following statement. For every integer m, 7m + 4 is not divisible by 7. Construct a…

A: See the solution below.

Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

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

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: Prove that 3n^2 + 1is divisible by 4 whenever n is an odd number.

A: Using induction method we have to prove that 3n^2 +1is divisible by 4 where n is odd number.

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

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Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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Q: For any positive integer n and r = 1,2, ...,n, we have G) = (","') + C" - }). True O False

A: The expression nr can be presented in the factorial form as follows. nr=n!(n-r)!r! The expression…

Q: 4. Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2+n², express an in terms of n. 5. Let, a1 = 1,…

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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…

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Q: Show that det(A) is (a* – b*)/(a – b), where a +b ab A = 1 a +b ab 1 a +b for a and b any real…

A: Given, A=a+bab01a+bab01a+b

Q: Show that if p q, then the formula: (3)* – 1 Ar fulfills the recursion formula - A =p. A+1 + q-…

A: As it is a multiple type question, according to the company policies, we can solve only the first…

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.