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# A manufacturer reckons that the value of a machine which costs him $Rs.15,625$, will depreciate each year by $20\%$. find the estimated value at the end of $5$ years.

$\begin{array}{1 1}Rs.3125 \\ Rs.5,120 \\ Rs.6,300 \\ Rs.5125 \end{array}$

Toolbox:
• $n^{th}$ term of a G.P. $=a.r^{n-1}$
Given that the cost of the machine $=Rs.15,625$.
The depreciation is $20$%. per year.
$\Rightarrow\:$ After one year the cost will be $15,625-\large\frac{20}{100}$$\times 15625 \qquad\qquad=15,625\big(1-\large\frac{1}{5}\big)$$=15,625\times(\large\frac{4}{5})$$=12,500 After two years the cost will be 12,500-\large\frac{20}{100}$$\times 12,500$$=12,500.\big.(1-\large\frac{1}{5}\big) \qquad\qquad\:=15,625\times\big(1-\large\frac{1}{5}\big)^2 The cost after 5 years will be 5^{th} term of the series 15,625\times\big(\large\frac{4}{5}\big) ,15,625\times\big(\large\frac{4}{5}\big)^2,......... which is a G.P. with first term a=15,625\times \large\frac{4}{5} and common ratio r=(1-\large\frac{1}{5})=\large\frac{4}{5} We know that n^{th} term of a G..P. =a.r^{n-1} \therefore t_5=15,625\times\large\frac{4}{5}\times (\large\frac{4}{5})^4$$=5120$
$i.e.,$ The cost of the machine after $5$ years will be $Rs.5,120$
edited Apr 11, 2014