Successive Compound Interest

Successive compound interest is a very powerful mathematical concept. This section introduces the concept of successive compound interest, and how it works in practice. This is a two-part article. The first part explains how compound interest works and the second part explains how it is calculated. In this article, we explain the concept of compound interest and its calculation. We also discuss the effect of compound interest on a person's lifetime income. We will also look at the mathematical successive compound interest formula, as well as its application in advanced mathematics.

Successive Compound Interest Formula

Here, we'll go through how to apply the variable rate of the compound interest formula. If the rate of compound interest for subsequent/consecutive years is different $(r_1 \%, r_2 \%, r_3 \%, r_4 \%$, etc.), then:

$\mathrm{A}=\mathrm{P}\left(1+\dfrac{r_1}{100}\right)\left(1+\dfrac{r_2}{100}\right) \quad\left(1+\dfrac{r_3}{100}\right)$

Where,

$r_1, r_2, r_3 \ldots \ldots . .=$ Rates for successive compound interest

$A=$ Amount

$\mathrm{P}=$ Principal

Amount in Compound Interest

The interest that is paid on both principal and interest and is compounded regularly is known as compound interest. The interest is calculated for the new principal at regular intervals when the accumulated interest and the current principal are combined. The initial principal plus the interest that has already accrued make up the new principal.

Compound Interest = Compounded Interest at Regular Intervals + Interest on Principal

The Formula of Amount in Compound Interest

Here mainly the formula of the amount in compound interest is given, which is as follows:

Compound Interest = Compounded Interest at Regular Intervals + Interest on Principal

$\mathrm{A}=\mathrm{P}\left(1+\dfrac{r}{n}\right)^{nt}$

Compound Interest $=\mathrm{P}\left(1+\dfrac{r}{n}\right)^{nt}-\mathrm{P}$

Here,

A = Amount in Compound Interest

P= Principal in Compound Interest

r= Rate of Compound Interest

n= Times interest Compounded annually

t = The overall tenure

Solved Examples of How to Find Rate in Compound Interest

Here are some examples, which mainly help us in understanding how to find the rate in compound interest in a better way:

Example- When the interest rate is 2% per year, what will the compound interest be on Rs. 2000 after two years compounded annually?

Ans: Given us, r= 2%, t= 2 years, P = 2000 rs.

$A=P\left(1+\dfrac{r}{n}\right)^{nt}$

$A=2000\left(1+\dfrac{2}{100}\right)^2$

A = 2000( 1.0404)

A= 2080.8 Rs

Compound Interest= A - P

= 2080.8- 2000

= 80.8 Rs

Thus the compound interest gained is 80.8 Rs.

Example- When the interest rate is 1% per year, what will the compound interest be on Rs. 1000 after 3 years?

Ans: Given us, r= 1%, t= 3 years, P= 1000 Rs.

$A=P\left(1+\dfrac{r}{n}\right)^{nt}$

$A=1000\left(1+\dfrac{1}{100}\right)^3$

A = 1000( 1.030301)

A= 1030.301 Rs.

Compound Interest= A - P

= 1030.301- 1000

=30.301 Rs

Thus the compound interest gained is 30.301 Rs.

Example- When the interest rate is 4 % per year, what will the compound interest be on Rs. 400 after 2 years compounded annually?

Ans: Given us, r= 4%, t= 2 years, P= 400 Rs.

$A=P\left(1+\dfrac{r}{n}\right)^{nt}$

$A=400\left(1+\dfrac{4}{100}\right)^2$

$A=400(1.04)^2$

$A=432.64 \mathrm{Rs}$

Compound Interest= A-P

= 432.64- 400

= 32.64 Rs

Thus the compound interest gained is 32.64 Rs.

Example- Mr. Kamal lent Rs. 2000 at compound interest of $10 \%$ per annum for 1 st year and $20 \%$ per annum for 2 nd year. What will be the total amount Mr. Kamal will have to pay after 2 years?

Ans: Given values, $P=R s .2000, R_1=10 \%, R_2=20 \%, n=2$ years, then

$A=P\left(1+\dfrac{R_1}{100}\right) \times\left(1+\dfrac{R_2}{100}\right)$

$A=2000\left(1+\dfrac{10}{100}\right) \times\left(1+\dfrac{20}{100}\right)$

$A=2000\left(\dfrac{11}{10}\right) \times\left(\dfrac{6}{5}\right)$

$A=2000 \times \dfrac{66}{50}=R s. 2640$

Practice Questions of Amount in Compound Interest

Here are some practice questions, which mainly help to understand successive compound interest formula in a better way:

Q1. When the interest rate is 1% per year, what will the compound interest be on Rs. 450 after 2 years compounded annually?

Ans: 9.045 Rs.

Q2. When the interest rate is 3% per year, what will the compound interest be on Rs. 950 after 1 year compounded annually?

Ans: 28.5 Rs.

Q3. When the interest rate is 5% per year, what will the compound interest be on Rs. 600 after 3 years compounded annually?

Ans: 94.575 Rs.

Q4. When the interest rate is 4% per year, what will the compound interest be on Rs. 1000 after 4 years compounded annually?

Ans: 169.858 Rs.

Q5. When the interest rate is 6% per year, what will the compound interest be on Rs. 920 after 2 years compounded annually?

Ans: 113.712 Rs.

Summary

Successive Compound interest is a very important concept in mathematics. It is the first of the three fundamental methods of financial mathematics and is used to calculate the growth rate of an amount. The topic of this chapter is compound interest, its calculation formula, and its applications in mathematics. This chapter also explains what compound interest is and how it is calculated and applied to mathematics. Some simple examples along with some practice questions, based on compound interest are discussed in this chapter.