Community Answer

【General guidance】The answer provided below has been developed in a clear step by step manner.Step1/3(a) The system of equations represents the amortization schedule of a mortgage with an initial loan amount of P and a monthly interest rate of r. Each equation in the system represents the remaining loan balance after making a certain number of payments. The two terms added in each equation represent the interest charged on the remaining balance and the payment made towards the loan principal. The lines are all equal because they represent the same remaining loan balance, just expressed in terms of different numbers of payments.Explanation:The lines are all equal because they represent the same remaining loan balance, just expressed in terms of different numbers of payments.Explanation:Please refer to solution in this step.Step2/3(b) We can use the system of equations to solve for \( \mathrm{{k}_{{{n}+{1}}}} \). Starting from the first equation, we have:\( \mathrm{{r}{P}+{k}_{{{1}}}={r}{\left({P}-{k}_{{{1}}}\right)}+{k}_{{{2}}}} \)Simplifying this equation gives:\( \mathrm{{k}_{{{1}}}={r}{\left({P}-{2}{k}_{{{1}}}\right)}+{k}_{{{2}}}} \)Substituting this into the second equation gives:\( \mathrm{{r}{\left({P}-{k}_{{{1}}}-{k}_{{{2}}}\right)}+{k}_{{{3}}}={r}{\left({P}-{k}_{{{1}}}-{2}{k}_{{{2}}}\right)}+{k}_{{{3}}}+{k}_{{{2}}}} \)Simplifying this equation gives:\( \mathrm{{k}_{{{2}}}={\left({1}+{r}\right)}{\left({P}-{k}_{{{1}}}\right)}-{k}_{{{3}}}} \)Substituting this into the third equation gives:\( \mathrm{{r}{\left({P}-{k}_{{{1}}}-{k}_{{{2}}}-{k}_{{{3}}}\right)}+{k}_{{{4}}}={r}{\left({P}-{k}_{{{1}}}-{2}{k}_{{{2}}}-{k}_{{{3}}}\right)}+{k}_{{{4}}}+{k}_{{{3}}}} \)Simplifying this equation gives:\( \mathrm{{k}_{{{3}}}={\left({1}+{r}\right)}{\left({P}-{k}_{{{1}}}-{k}_{{{2}}}\right)}-{k}_{{{4}}}} \)Continuing in this way, we can derive the following expressi ... See the full answer