Year: 2002
Paper: 1
Question Number: 8
Course: LFM Pure and Mechanics
Section: Arithmetic and Geometric sequences
No solution available for this problem.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1516.0
Banger Comparisons: 1
I borrow $C$ pounds at interest rate $100\alpha \,\%$ per year.
The interest is added at the end of each year. Immediately after the
interest is added, I make
a repayment. The amount I repay at the end of the $k$th year is $R_k$
pounds and the amount I owe
at the beginning of $k$th year is $C_k$ pounds (with $C_1=C$).
Express $C_{n+1}$ in terms of $R_k$ ($k= 1$, $2$, $\ldots$, $n$), $\alpha$ and $C$
and show that, if I pay off the loan in $N$ years with repayments
given by $R_k= (1+\alpha)^kr\,$, where $r$ is constant, then $r=C/N\,$.
If instead I pay off the loan in $N$ years with $N$ equal repayments
of $R$ pounds, show that
\[
\frac R C = \frac{\alpha (1+\alpha)^{N} }{(1+\alpha)^N-1} \;,
\]
and that $R/C\approx 27/103$ in the case $\alpha =1/50$, $N=4\,$.