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Q.
A manufacturer reckons that the value of a machine, which costs him ₹ 15625 , will depreciate each year by $20 \%$. The estimated value at the end of $5 yr$ is
Sequences and Series
Solution:
At present cost of machine $=₹ 15625$
Now, after $ 1 $ yr cost of machine $ =15625-\frac{20}{100} \times 15625 $
$ =15625-3125 $
$ -₹ 12500$
After $ 2 yr$ cost of machine $ =12500-\frac{20}{100} \times 12500$
$ =12500-2500=₹ 10000$
and so on.
Clearly, the cost of machine form a G.P. 15625, 12500 , $10000, \ldots$ with first term $a=15625$, common ratio $(r)=\frac{4}{5}$
Now, the estimated value at the end of $5 yr$ is given by
$T_6 =a r^5=15625\left(\frac{4}{5}\right)^5 $
$ =15625 \times \frac{4^5}{5^5}=5^6 \times \frac{2^{10}}{5^5}$
$ =5 \times 1024=₹ 5120$
Note In depreciation, the cost value decreases every year.