$0<r<1$ ($100r$ is the annual percentage rate). The bank charges interest every month. You take a loan of $L$ pounds and wish to pay it back in exactly $m$ months by making monthly payments of $p$ pounds each. Find $p$.
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- Evaluate the sum
$a+ar+ar^2+\cdots+ar^n,$with$r \neq 0$and$n>0.$ - Given
$a_0= 5$and$a_n= 3a_{n-1}+4,$find$a_4.$
- Evaluate the sum
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Find an expression for the amount remaining to be paid after month
$i.$ -
… in terms of the amount remaining to be paid in month
$i-1.$ -
You might first want to find the cumulative monthly interest rate. Hint: it’s not
$r/12.$ -
… knowing that the cumulative monthly interest rate is the interest rate that gives
$r$when applied successively$12$times. -
If
$x_i$denotes the amount remaining to be paid after month$i,$you should now have a recursive formula for$x_i.$Try to find the explicit formula. -
You should get an expression that contains
$x_0.$What is the value of$x_0?$
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Let
$x_i$be the amount remaining to be paid after month$i$. Thus,$x_0=L.$We have$x_i=x_{i-1}(1+\mu)-p,$where$\mu$is the cumulative monthly interest rate (we’ll determine its value later). To simplify our working, let$c=1+\mu.$Unwinding the recursion, we get:$$ \begin{aligned} x_i &= x_{i-1}c-p \\ &= (x_{i-2}c-p)c - p \\ &= \left(\left((x_{i-3}c-p)c - p\right) \right)c-p \\ &= \ldots \\ &= x_0c^i-p\sum_{k=0}^{i-1}c^k \\ &= Lc^i-p\frac{c^i-1}{c-1}. \end{aligned} $$We want
$x_m=0,$which gives$p=L\;c^m\;\frac{c-1}{c^m-1}.$Now we need to express
$\mu$in terms of$r.$The cumulative monthly interest rate$\mu$is the interest rate that gives$r$when applied successively (i.e. recursively)$12$times. This is not equal to$r/12\;^{(*)}.$An initial amount$z_0$becomes$z_0(1+\mu)$after a month, and$z_0(1+\mu)^{12}$after$12$months, which must also be equal to$z_0(1+r).$This is in fact another recursion:$z_i=z_{i-1}(1+\mu),$where we have$z_{12}=z_0(1+\mu)^{12}=z_0(1+r).$Hence,$\mu=(1+r)^{1/12}-1.$The monthly payment sought is then
$p=L\;(1+r)^{m/12}\;\frac{(1+r)^{1/12}-1}{(1+r)^{m/12}-1}.$(*)
$\mu$can in fact be approximated by$r/12$for$r\ll1$though Taylor Series. This is a safe assumption for savings accounts, but not for loans (hmm, what does that tell you about banks?).