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Home » 3k = -7 \Rightarrow k = -\frac73 - Get link 4share

3k = -7 \Rightarrow k = -\frac73 - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 So, \( x = 2 \) is a root. Use synthetic division to divide \( m(x) \) by \( x - 2 \):

📌 & 2 & -9 & 12 & -4 \

📌 & & 4 & -10 & 4 \

📌 & 2 & -5 & 2 & 0 \

📌 The quotient is \( 2x^2 - 5x + 2 \). Solve \( 2x^2 - 5x + 2 = 0 \) using the quadratic formula:

📌 x = rac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} = rac{5 \pm \sqrt{25 - 16}}{4} = rac{5 \pm \sqrt{9}}{4} = rac{5 \pm 3}{

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Multiplying through by \(k(k+1)\) gives:
= 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:
Thus, the sum is:

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