This is the formula for the present value of an annuity. We can find R$,$ given P $,$ n and i

as

R$=$Pi$.1(1+$i$)$n

In R$,$ we define variables as follows: principal to hold the value of P $,$ and intRate to hold the interest rate, and n to hold the number of payments. We will assign the resulting payment value to an object called payment.

Of course, we need some numerical values to work with, so we will suppose that the loan amount is $$1500,$ the interest rate is $1\%$ and the number of payments is $10.$ The required code is then

intRate $<-0\mathrm{.}01$

n $<-10$

principal $<-1500$

payment $<-$ principal $*$ intRate $/(1-(1+$ intRate$)(-$n$))$

payment

## $[1]158\mathrm{.}3731$

For this particular loan, the monthly payments are $$158\mathrm{.}37.$

$1$

In each question below, write out $($or type$)$ the required lines of R code, together with the answer to the question.

$1.$ Calculate the monthly payment required for a loan of $$200,000,$ at a monthly interest rate of $0\mathrm{.}003,$ based on $300$ monthly payments, starting in one month$$s time.

$2.$ Calculate the sum $\backslash$Sigma from n to j$=1$ of j and compare with n$($n $+1)/2,$ for n $=100,200,400,800.$