Question

A person has deposited Rs. 25,000 in an investment which yields 14% simple interest annually. Do these amounts (principal + interest) from an A.P.? if so, determine the amount of investment after 20 years.

Answer 1

Hint:In the solution, first we have to find the interest amount. For that we use the simple interest formula $I = Ptr$, where $P$ is the principal amount, $t$ is the time in years and $r$ is the percentage rate. Since, this amount per year represents a series of A.P., so the simple interest $I$ represents the common difference of A.P. From that we have to calculate the amount next year. This will represent the first term of A.P. Once we get the first term and the common difference we can calculate the ${20^{th}}$ term using the formula${a_n} = {a_1} + (n - 1)d$, where ${a_1}$ is the first term of an A.P., $n$ is the number of terms, $d$ is the common difference and ${a_n}$ is the ${n^{th}}$ term of the A.P. series.

Complete step by step solution:
Given that the principal amount is Rs. 25,000, rate 14%.

Interest
$\begin{array}{c}I = Ptr\\ = 25000 \times 1 \times \dfrac{{14}}{{100}}\\ = 25000 \times 1 \times 0.14\\ = 3500\end{array}$
Since, this amount forms an A.P.
Therefore, the common difference is Rs.3500.

After one-year amount will be = Rs. 25,000 + Rs.3500 = Rs. 28,500

Thus, the first number in the series of A.P. is Rs. 28,500.
It is known the ${n^{th}}$ term of an A.P. is ${a_n} = {a_1} + (n - 1)d$, where ${a_1}$ is the first term of an A.P., $n$ is the number of terms, $d$ is the common difference and ${a_n}$ is the ${n^{th}}$ term of the A.P. series.
Now, substituting 20 for $n$, Rs.28,000 for ${a_1}$ and Rs.3500 for $d$.
Therefore, investment amount after 20 years is
$\begin{array}{c}{a_{20}} = {a_1} + (n - 1)d\\ = {\rm{Rs}}.28000 + \left( {20 - 1} \right) \times {\rm{Rs}}.3500\\ = {\rm{Rs}}.95,000\end{array}$

Hence, the investment amount after 20 years is Rs. 95,000.

Note: Simple interest is calculated on the basis of a basic amount borrowed for the entire period at a particular rate of interest and arithmetic progression (A.P.) is a series where each term is equal to the sum of its previous term and a fixed term. Here we have to determine the amount of investment after 20 years. Since the initial amount and the rate of interest is given, we can easily calculate the amount after 20 years by substituting the values in the formulas.