The price of a new car is $RM80000$ . It is given that the price of the car depreciates at a constant rate if 5% yearly. Calculate the minimum number of years required for the price of the car to drop to less than $RM45000$ .?

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USE FORMULA : $T_n = ar^(n-1)$

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Answer 1 · 2018-04-16T09:46:44

$color(blue)(13)$

$ar^(n-1)$

This is just the nth term of a geometric sequence.

If the car is depreciating at $5%$ yearly, then we can say that the value of the car after each year is $95%$ of its value the previous year.

Using this as our common ratio and expressing it in decimal form, and using $a$ as our initial value, the nth term will be:

$80000(0.95)^(n-1)$

We need this to be less than 45000. First we solve for $n$ equal to this amount:

$80000(0.95)^(n-1)=45000$

Dividing by $80000$

$(0.95)^(n-1)=45000/80000=45/80=9/16$

Taking natural logarithms of both sides:

$(n-1)ln(0.95)=ln(9/16)$

$n-1=ln(9/16)/ln(0.95)$

$n=ln(9/16)/ln(0.95)+1~~12.21714158$

This is for equality, and we also need an integer value. We therefore go to the nearest integer greater than this:

$n=13$

So:

$T_13=80000(0.95)^(13-1)=43228.80702$