Let -6 M = Find formulas for the entries of M", where n is a positive integer.

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: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: Find an expression for An, where n is an arbitrary positive integer.

A: Given: A=1423∈M2×2R

Q: 2. Find an explicit formula for аn 3 За,-1+ 10ар-2 and ao %3D 0, ај 3D 7 An where

A: The solution is given as

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Q: 4. Use the PMI to prove the following for all natural numbers n. |(c) E-1 2' 2n+1 – 2 i=D1

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Q: Find all integer solutions 2² – 2y? = 22 satisfying (x, y, z) = 1.

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Q: If A = (; A-: )-..R. e M2x 2(R), find A" for any positive integer n.

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

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Q: 3. Let, a, = 3, a2 = 4 and for n2 3, a, = 2an-1+ an-2 + nº+ 1, express a, in terms of n.

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Q: Prove 1 +,+ i++ = 2 – for all positive integers n. |

A: We want to prove 1 + 12 + 14 + ... + 12n = 2 - 12n

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

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

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

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Q: M d formulas for the entries of M", where n is a positive integer.

A: To find the power of a matrix

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A: Consider that x is a random variable. From the formula of population variance;

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: Let M = 6 1 -2 9 Find formulas for the entries of M", where n is a positive integer. Mn -

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Q: The small positive integer n for which (1+i)?"=(1-i)?n is =..

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Q: 4. Show that for all formulas p, p,0 E S(P), (a) (9 = 4) = ¬(4 ^ ¬ab).

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Q: 4) if n> 1, then (1+x)" > 1+nx for x E R+ (i.e. x is a non-negative real number). MATH151

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

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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: b) Prove that () - 2) = n², where n is a positive integer.

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Q: -a = Y express L and K as expressions of

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Q: 4. For each positive integer i, let A, = {x € R| -<x < } = (-;;4) a. Find U-, A;. i=1 b. Find N 4:-…

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Q: a + b (2) (a) Prove by contrapositive : Va, b e R* : a <b → a < < b. (b) Prove by contradiction :…

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Q: of t/4 for x. (If an answer is undefined, enter UNDEFINED.)

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Q: 3. Let, a1 = 3, a2 = 4 and for n 2 3, an = 2an-1 + an-2 -5, express a, in terms of n. %3D

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Q: 5) Let A= 18 Using Caley-Hamilton theorem find the expression A* -19A³ + 18A? + A + 21.

A: Please refer the attached image for complete solution.

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

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

A: We will solve the problem using recurrence relation

Q: For every integer n23 I+ 2+ 2?+ +4^= 4(4n-16) %3D 3. Proof (by induction)

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Q: How many nonnegative integers a, b, c.2 and d are there that satisfy the equation 1 + + a+b 1 a…

A: Given equation has 4 variables a,b,c,d Sum=1 We can find pairs by the following ways

Q: Q2/ prove [A] [A] = [I] if [2 4 -5 [A]= 2 7 1 3 4

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Q: Let...

A: Given expansion formula is  a+bn=∑nkk=0nan-kbk

Q: 5 М — 11 Find formulas for the entries of M", wheren is a positive integer. M" =

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Q: Let M=( 3 Find formulas for the entries of M", where n is a positive integer. Mn = =

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Q: 5. For non-negative integer n and r, where 1<rsn, show that (n-r+1)"C,1=n(""C,). r-1

A: Given n and r non negative integers with  1≤ r≤ n. We need to prove (n-r+1)nCr-1=n(n-1Cr-1)

Q: Let i be a positive integer and A, = [-i+1,1[. Then N14= %3D

A: We have to find the first intersection first.

Q: Let n be a positive integer and M %3D Then M"

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Q: Find integers s and t so that s· 201 + t · 302 = 1 . S %3D

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Q: Q4/ If A = 3 - l: prove A-A-5I = 0

A: Given that: A=213-1

Q: a + b (2) (a) Prove by contrapositive : Va, b e R* : a < b → a <• (b) Prove by contradiction : For…

A: Part(a) We need to prove by contrapositive. In proof by contrapositive, if we have a statement "if p…

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

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Q: Let -2 М- -16 Find formulas for the entries of M", wheren is a positive integer. M" =

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Q: Let М- Find formulas for the entries of M", where n is a positive integer. M" =

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