A farmer buys a used tractor for $Rs$ $12000 .$ He pays $Rs$ $6000$ cash and agrees to pay the balance in annual instalments of $Rs$ $500$ plus $12 \%$ interest on the unpaid amount. How much will the tractor cost him?

It is given farmer pays $Rs.$ $6000$ in cash.

Therefore, unpaid amount $=$ $Rs.$ $12000-$ $Rs.$ $6000=$ $Rs.$ $6000$

According to the given condition, the interest paid annually is

$12 \%$ of $6000,12 \%$ of $5500,12 \%$ of $5000 \ldots \ldots 12 \%$ of $500$

Thus, total interest to be paid

$=12 \%$ of $6000+12 \%$ of $5500+12 \%$ of $5000+\ldots \ldots+12 \%$ of $500$

$=12 \%$ of $(6000+5500+5000+\ldots .+500)$

$=12 \%$ of $(500+1000+1500+\ldots \ldots+6000)$

Now, the series $500,1000,1500 \ldots 6000$ is an $A.P.$ with both the first term and common difference equal to $500 .$

Let the number of terms of the $A.P.$ be $n$

$\therefore 6000=500+(n-1) 500$

$\Rightarrow 1+(n-1)=12$

$\Rightarrow n=12$

$\therefore$ Sum of the $A.P.$

$=\frac{12}{2}[2(500)+(12-1)(500)]=6[1000+5500]=6(6500)=39000$

Thus, total interest to be paid

$=12 \%$ of $(500+1000+1500+\ldots . .+6000)$

$=12 \%$ of $39000= Rs .4680$

Thus, cost of tractor $=( Rs .12000+ Rs .4680)= Rs .16680$