2. Let an = 3n + 2 for n = 0, 1, 2, ... (i ) Find ao, a1 and a2. (ii ) Show that a2 = 4a1 – 3ao. (iii ) Show that an = 4an-1 - 3an-2 for all integers n with n > 2.

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A: We have to prove given identity:

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A: Given :              n2+6n+11 = x2+y2              t ≤ t

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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: (b) Define fo = 0, f₁ = 1 and fn = fn-1 + fn-2 for n ≥2. Show that 3 fak for all k ≥ 0.

A: As per the guidelines we shall solve one question only as you have asked more than one different…

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Q: a. Prove that the v63 iş irrational b. For all integers, prove that 4 (n2 + 2) c. Prove or disprove:…

A: Note: As per our company guidelines we are supposed to answer the first three sub-parts of the…

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Q: Compute 4n — бп? lim Vlan| for a,n %3D 3n2 +7 n-00

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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. Let, a, = 1, a2 = 2, b, = 0, b2 = 1, for n2 3, an = 2an-1+ bn-1 for n22 b, = bn-1+ an-1 express…

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

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Q: 8a, 2+ 225n for n> 2 with initial conditions ao 7, a1 83 %3D

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Q: Show that any positive integer n with n ≥ 4 can be expressed as n = 2x + 5y where x ≥ 0, y ≥ 0.

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Q: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

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Q: Prove that: 3n > 2n + n2 for all integers n is greater than or equal to 2

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Q: 8. Prove that n Ei i! = (n+1)! – 1 i=1 for all positive integers n.

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Q: Prove T(n)= 2T(n-1)+n can be in 0(n2")using represented telescoping method

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Q: Show that for all n E Z, n² 5.

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A: Given: For all n∈ℤ, n2-3n+1≠2

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Q: prove using induction: II ( 1 1 1 П 2k +1 2k+2, for n 20 (2n + 2)! k-0

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Q: Prove that gn = 5 · 3n + 7 · 2n for all integers n ≥ 0.

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

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Q: Prove that: "C. = 2" {1.3.5...(2n - 1)} n!

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Q: n Prove that (3i – 2) = ; (3n – 1) for all integers n > 1. - i=1

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Q: part ii and iii, thank you

A: According to pascal law,

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

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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: . Let, a, = 1, az = 2, b; = 0, b2 = 1, for n2 3, a, = 2an-1 + bn-1 for n22 b, = br-1+ an-1 express…

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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: 2n° 1. Prove that for all positive integers n > 1, the inequality 2" < <(") holds. п

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Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

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

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Q: 2 Lk=0 4. Prove that for all positive integers n, п

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Q: 8

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