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Home » Now, we solve this system of equations. Start with the first two equations: - Silent Sales Machine

Now, we solve this system of equations. Start with the first two equations: - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 = A(k+1) + Bk = Ak + A + Bk

📌 = (A + B)k + A

📌 For the equation to hold for all \(k\), we set up the system:

📌 A + B = 0 \quad \text{and} \quad A = 1

📌 Solving this, we find \(A = 1\) and \(B = -1\). Thus:

📌 Now, substitute this into the sum:

📌 This is a telescoping series, where most terms cancel out:

📌 After cancellation, we are left with:

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We write:
u^4 + 3 = (u^2 - 2u + 2)Q(u) + au + b
To find \( a \) and \( b \), perform polynomial division or use substi...
Letâs use the roots of \( u^2 - 2u + 2 = 0 \). The discriminant is \...
u = rac{2 \pm \sqrt{-4}}{2} = rac{2 \pm 2i}{2} = 1 \pm i
Let \( lpha = 1 + i \), \( eta = 1 - i \). Then at \( u = lpha \):
lpha^4 + 3 = alpha + b
First compute \( lpha^2 \):
lpha = 1 + i \Rightarrow lpha^2 = (1+i)^2 = 1 + 2i + i^2 = 1 + 2i -...
lpha^4 = (lpha^2)^2 = (2i)^2 = 4i^2 = -4

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