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Home » Label Address Label Hacks That Every Professional Swears By—Click to Master It! - Carbonext

Label Address Label Hacks That Every Professional Swears By—Click to Master It! - Carbonext

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 \( bc + a^2 = 1 \)

📌 Both equations are identical. So, \( a^2 + bc = 1 \) holds. Thus, the conditions are \( d = -a \) and \( a^2 + bc = 1 \).

📌 The valid matrices \( T \) are:

📌 T = egin{pmatrix} a & b \ c & -a \end{pmatrix} \quad ext{with} \quad a^2 + bc = 1

📌 oxed{egin{cases} T = egin{pmatrix} a & b \ c & -a \end{pmatrix}, & a^2 + bc = 1 \end{cases}}

📌 Question:** A mathematician working in applied mathematics at a financial institution models compound interest with a discrete-time model. Suppose an investment grows according to the recurrence \( A_{n+1} = r A_n \) with \( A_0 = P_0 \). If \( A_5 = 3P_0 \), find the annual growth rate \( r \) and the smallest integer \( n \) such that \( A_n > 10P_0 \).

📌 Solution:** From the recurrence \( A_{n} = r^n P_0 \), and given \( A_5 = 3P_0 \), we have:

📌 r^5 P_0 = 3P_0 \Rightarrow r^5 = 3 \Rightarrow r = 3^{1/5}

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