А = 950 тысяч рублей кредит сроком на (n+2) месяца под 2% в месяц.
| Месяц |
Начисленные % |
Остаток |
| 1 |
\(950 \cdot \frac{2}{{100}}\) |
\(950 — a\) |
| 2 |
\(\)\(\left( {950 — a} \right)\frac{2}{{100}}\) |
\(950 — 2a\) |
| … |
… |
… |
| n |
\(\left( {950 — a\left( {n — 1} \right)} \right)\frac{2}{{100}}\) |
\(950 — an\) |
| n+1 |
\(\left( {950 — an} \right) \cdot \frac{2}{{100}}\) |
\(950 — an — 300\) |
| n+2 |
\(\left( {950 — an — 300} \right) \cdot \frac{2}{{100}}\) |
\(950 — an — 600 = 0\) |
Так как сумма кредита 950 тысяч рублей, а общая сумма выплат 1188,5 тыс. рублей, то переплата, то есть начисленные проценты составляют: \(1188,5 — 950 = 238,5\) тыс. рублей.
\(\left\{ {\begin{array}{*{20}{c}}{950 \cdot \frac{2}{{100}} + \left( {950 — a} \right)\frac{2}{{100}} + … + \left( {950 — a\left( {n — 1} \right)} \right) \cdot \frac{2}{{100}} + \left( {950 — an} \right) \cdot \frac{2}{{100}} + \left( {950 — an — 300} \right) \cdot \frac{2}{{100}} = 238,5}\\{950 — a\,n — 600 = 0.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\end{array}} \right.\)
Из второго уравнения: \(a\,n = 350.\) Из первого уравнения:
\(\frac{2}{{100}} \cdot \left( {\frac{{950 + 950 — an}}{2} \cdot \left( {n + 1} \right) + 300} \right) = 238,5\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\frac{{1900 — 350}}{2} \cdot \left( {n + 1} \right) = 11925 — 300\,\,\,\,\, \Leftrightarrow \,\,\,\,n = 14.\)
Ответ: 14.