If P(n) is the statement "n' + n is divisible by 3' Prove that P(3) is true but P(4) is not true.

Q: 1), n = 1, 2, . .

A: In this question, the concept of mathematical induction is applied. Mathematical Induction Inductive…

Q: Why is it that if I do row expansion on the first row, I get det(A) = -1, but, if I do it in the…

A: We need to find the determinant of a using the first row expansion and as well using the second…

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

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

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Q: D. Show that if a, = 3n³ - 17n² + 2n - 6, then an O(n³).

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Q: n2 +2 2 3n for n = 1,2,3,.

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Q: 4. Conclude that n2 n=1 ||

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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: 4. Find the coefficient of the term containing x' in the expansion of (x-2y)“.

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Q: Prove that S4 = ((1 2 3 4) , (1 2 4 3) ). %3D

A: As you asked for question number 8 , and hence here it is.

Q: 4. Find the expansion of (x + y)6 using Binomial Theorem.

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Q: Suppose we have L-shaped tiles made up of three squares: Prove that for all n > 1, it's possible to…

A: Suppose we have L shaped tiles Prove that for all n>=1 it's possible to lay these tiles down to…

Q: Is it true that if a, < b, for all nE N and E-1 a, is divergent, then E-1 bn is divergent? If false…

A: Given: an≤bn for all n∈ℕ To check that if ∑n=1∞an is divergent then ∑n=1∞bn is divergent

Q: If (n + 1)(n + 2)an+1x" - nan-1x" = 0, n=1 show that for k = 1, 2, 3, ..., we have 1.3.5... (2k - 1)…

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Q: 4. With the fact of 1² +2²+...+ n² = n(n+1)(2n + 1)/6, prove that n n n(n −1) ΣΣ(i-j)² 6 i=1 j=1…

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

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Q: D. Show that if an 3n3 - 17n2 + 2n - 6, then a, = O(n³).

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Q: Which term of the expansion of (z + 4) contains z Term number contains z4.

A: Given expression is: (x4+4)16

Q: Proceed as in Example 3 in Section 6.1 to rewrite the głven expression using a single power serles…

A: Given ∑n=2∞nn−1cnxn−2+∑n=0∞cnxn+2 We have to rewrite the given expression using a single power…

Q: a)Show by one combinatorial method that: 2 2 n C) - () - (") 2n +... + 2 n

A: Given information: The following identity is provided: n02+n12+n22+................+nn2=2nn It is…

Q: Proceed as in Example 3 in Section 6.1 to rewrite the given expression using a single power n(n –…

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Q: Prove bY inductions (R4

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Q: Show that 2n+1≤2^n  for all n≥3.

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

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Q: Prove n (2n = n n 1 k=1

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Q: 1. Prove that {2" : n E Z} = {,n :n E Z}.

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Q: Prove that Vn 2 2: 1++÷+ …+ 1 1 1 < 2 – ... 4 9 n2 n

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Q: Show by two methods one algebraic and one combinatorial that: ΣΗ (E) (-) = (m*") -k=0

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Q: 2. Prove by the Principle of Mathematical Induction that 5-2 + 5-4 + 5-6 +... + 5-2n = 24-'(1 –…

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Q: 15. Find the general solution of dr bcosedr + rsinode

A: General solution of differential equation

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

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Q: Hugh was trying to see if he could tile a 2" x 2" board with L-shaped tiles (i.e. a 2 X 2 tile with…

A: Given: We have given a 2n×2n board and we want to fit the board with L-shaped tiles such that ∃ an…

Q: 6. Show directly that inf(B) = 1 for B {N =: 4+1} -

A: The given expression is,  B = {1+1/n : n belong to N}

Q: Consider the following LP Min z= 3x, + 2.x, +7x, s.t - x, +x, =10 -2x, + x, - x, <-10 X,,X,,X, 2 0…

A: In the given LP problem when we solve it via simplex method, we use a series of tableaus.

Q: If (ar + bx) = 4n and n+1 k-1 (ak-br) = then ar = n+2 k=1 k-1

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Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: 2. Show that for every n 2 2 and every 0 <k<n- 2, within the triangle of Binomial Coefficients k+2…

A: We will use properties of binomial coefficient  nCr  +  nCr-1=n+1Cr+1Note : nCr=nr=n!r! n-r!   Proof…

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

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

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

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Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

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Q: if gt2) Use induction to prove: 1 ΣΤΗ Σ 2 2n n > 0. hint: 1 2n 2 2n+1

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Q: 6. Show that (") = 3() + 6n (5) +n³. Using Combinatorial Proof.

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Q: 6. Prove that for n> 2, 2(;) + (*) =n². п

A: We have too prove the given expression.   According to our guidelines we can solve only one question…

Q: A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second day, he…

A: Given - A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second…

Q: If s(n)=3n2−3n+5s(n)=3n2−3n+5, then s(n)=2s(n−1)−s(n−2)+cs(n)=2s(n−1)−s(n−2)+c for all integers…

A: Function represents the relation between dependent variable and independent variable. Function…

Q: 8. Working over R, suppose that M = PDP ' where P 1 Then, for any positive integer k, we have that…

A: To find Mk =?

Q: What is the solution to the recurrence relation: Tn=5nTn-1+6 with T0=3?

A: We will find out the required solution.