Question

The population of a town increases by $5\%$ every year. If the present population is 5,40,000, find the population after 2 years.
[a] 5,95,350
[b] 5,94,000
[c] 5,95,000
[d] 5,94,350

Answer 1

Hint: Find the population of the town after 1 year which would be $5\%$ more than what is currently. Then find the population of the town after two years using population after one year. Alternatively derive a formula for population after n years and put n = 2. Use the property that if $\left\{ {{a}_{n}} \right\}$ is a G.P of common difference r then $\dfrac{{{a}_{n+1}}}{{{a}_{n}}}=r$ and ${{a}_{n}}={{a}_{m}}{{r}^{n-m}}$.

Complete step-by-step answer:
Here Current Population = 5,40,000.
So, population after one year $=5,40,000\text{ }+\text{ }5\%\text{ }of\text{ }5,40,000$
$\begin{align}
  & =540000+\dfrac{5}{100}\times 540000=540000=540000+27000 \\
 & =567000 \\
\end{align}$
Population after 1 year = 5,67,000.
So, population after two years $=5,67,000+\text{ }5\%\text{ }of\text{ }5,67,000$
$\begin{align}
  & =567000+\dfrac{5}{100}\times 567000 \\
 & =567000+28350 \\
 & =595350 \\
\end{align}$
Hence the population after two years = 5,95,350.
Hence option [a] is correct.

Note:Alternative Method:
Let ${{P}_{n}}$ be the population after n years.
So, according to the question
$\begin{align}
  & {{P}_{n+1}}={{P}_{n}}+0.05{{P}_{n}} \\
 & \Rightarrow \dfrac{{{P}_{n+1}}}{{{P}_{n}}}=1.05 \\
\end{align}$
which is a geometric series with r = 1.05 and $a={{P}_{0}}$ because the ratio between adjacent terms is constant and the ratio is equal to 1.05.
Hence ${{P}_{n}}={{P}_{0}}{{\left( 1.05 \right)}^{n}}$. Observe that we wrote ${{\left( 1.05 \right)}^{n}}$ and not ${{\left( 1.05 \right)}^{n-1}}$ even though the formula for ${{a}_{n}}$ is ${{a}_{n}}=a{{r}^{n-1}}$. This is because here we are starting our count from 0 and not from 1. You can also use the formula ${{a}_{n}}={{a}_{m}}{{r}^{n-m}}$. Put m = 0 to get the above result.
Substituting ${{P}_{0}}=540000$and n = 2, we get
${{P}_{2}}=540000{{\left( 1.05 \right)}^{2}}=540000\times 1.1025=595350$
Hence the population after years = 5,95,350
Hence option [a] is correct