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.

Hint: Take the income of the person in the first year as ‘a’ and the amount by which it increases every year as ‘d’. Now the salary of each year would form A.P. Find the total amount he received in 20 years by using the formula for the sum of n terms of A.P that is ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d\right]$

Complete step by step solution:

Here we are given that the income of a person is Rs.300,000 in the first year and it increases every year for the next 19 years by Rs.10000. We have to find the total amount he received in 20 years.
Let us consider the income of the person in the first year as
a = Rs.300,000

Also, let us consider the amount by which the income is getting increased every year as d = Rs.10000.
So, we get the income of a man in the first year = a.
Also, the income of a man in the second year = a + d.
Similarly, the income of a man in the third year = a + 2d.
This would continue for a total of 20 years.
So, we get the series of income of a man in each year as
a, a + d, a + 2d, a + 3d………
Here, we can see the income of a man in each year in A.P with a = Rs.300,000 as first term and d = Rs.10000 as a common difference.
We know that sum of n terms of A.P $=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]....\left( i \right)$
Now, we have to find the total amount he received in 20 years. So, we have to add his income from the first year to the 20 th year. As we know that his income is in A.P. So, we get
Total amount received by man in 20 years = Sum of 20 terms of A.P $\left( {{S}_{20}} \right)....\left( ii \right)$
By substituting the value of n = 20, a = 300,000 and d = 10000 in equation (i), we get,
Sum of 20 terms of A.P $=\left( {{S}_{20}} \right)=\dfrac{20}{2}\left[ 2\times \left( 300000 \right)+\left( 20-1 \right)\left( 10000 \right) \right]$
${{S}_{20}}=10\left( 600000+190000 \right)$
$=10\left( 790000 \right)$
$=79,00,000$
By substituting the value of ${{S}_{20}}$ in equation (ii), we get,
Total amount received by man in 20 years = Rs.7900000
So, we get the total amount received by man in 20 years as Rs 79 lakhs

Note: Here, some students try to manually find the total amount by adding the income of each year one by one. But this method is very lengthy and can even give wrong results if there would be even a slight mistake in calculation. So students must identify the series in these types of questions and accordingly use the formula of the sum of n terms.