Question

The income of a person is Rs. 300,000 in the first year and he receives an increase of Rs. 10,000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.

Answer 1

Hint: The person’s income is increasing every year at a rate of Rs. 10,000 every year. This can be formed as an Arithmetic progression. Here the 1st term a will be 300,000 and the common difference d will be equal to 10,000. We, can find the total amount for 20 years by taking the sum of 1st 20 terms of the AP using the equation ${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$

Complete step by step Answer:

The person’s 1st year salary is Rs. 300,000. Every year the salary increases by Rs.10,000. This can be formed as an arithmetic progression with first term \[{\text{a = 300,000}}\]and common difference \[{\text{d = 10,000}}\]
The AP can be written as 300000, 310000, 320000, ……
The sum of 1st n terms of an AP is given by,
${S_n} = \dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ where a is the 1st term, d is the common difference and n is the number of terms.
We need to find the sum of the salary of 20 years. The total amount received by the person in 20 years is given by the sum of the 1st 20 terms. So, we can take n as 20.
On substituting values of a, d, and n, we get,
$
  {S_{20}} = \dfrac{{20}}{2}\left( {2 \times 300000 + \left( {19} \right) \times 10000} \right) \\
   = 10\left( {600000 + 190000} \right) \\
   = 10 \times 790000 \\
   = 79,00,000 \\
 $
Therefore, the amount earned in 20 years is Rs. 79,00,000.

Note: The concept used here is Arithmetic Progression. The formula for calculating the sum up to n terms must be known. As the figures are too large there is a chance of making mistakes, so always recheck after calculating. For an AP with 1st term a and common difference d, the nth term is given by the equation, ${a_n} = a + \left( {n - 1} \right)d$ and the AP is written as $a,a + d,a + 2d,....,a + \left( {n - 1} \right)d$. We can find any term of an AP if we know the value of a and d. We recognized this as an AP as the common difference is added to each term. If a common number is multiplied to each term, then the series will become a GP or geometric progression. The common number which is multiplied is known as the common ratio.