Question

The income of a person is Rs. 300,000 in the first year and he receives an increase of Rs. 10000 to his income per year for the next 19 years. Find the total amount he received in 20 years.

Answer 1

We will form an A.P. based on the given conditions. The A.P. will have a first term as 300,000 and the common difference will be 10,000. We have to find the total amount in 20 years, therefore, the value of $n = 20$. Then, apply the formula of sum of A.P. , $\dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$.

Complete step by step solution:
The income of the person is Rs. 300,000 in the first year.
The increase in the person's income per year is given as Rs. 10000
Then, the income of the person in second year will be $300,000 + 10,000$
Similarly, the income in third year will be $300,000 + 2\left( {10,000} \right)$
If we continue like this, we will get an A.P
$a,a + d,a + 2d....a + \left( {n - 1} \right)d$, where $a = 300,000$, $d = 10,000$ and $n = 20$
We want to find the total amount received in 20 years.
We will use the formula to find the sum of $n$ terms, $\dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$
On substituting the values of $a,d,n$ in the above formula, we will get.
$\Rightarrow \dfrac{{20}}{2}\left( {2\left( {300,000} \right) + \left( {19} \right)10,000} \right) \\\ \Rightarrow 10\left( {600,000 + 190,000} \right) \\\ \Rightarrow 7,900,000 \\\$

Thus, the total amount received in 20 years is ${\text{Rs}}{\text{. }}7,900,000$

Note:
We can also calculate the term using the formula, ${S_n} = \dfrac{n}{2}\left( {a + {a_n}} \right)$, where ${a_n}$ is the last term and is calculated using the formula, ${a_n} = a + \left( {n - 1} \right)d$, where $a$ is the first term and $d$ is the common difference of the A.P.