Numerical entities can either grow or reduce with time. For e.g., age of living and non living things increase with time, radioactivity of radioactive substances decrease with time,
bank balance increases (or decreases) with time etc. There are two aspects associated with the process of growth or reduction.

- Growth or reduction happens on some initial value of entity. This value is known as
.*Principal* - Growth or reduction happens
some value. For process of growth this value is known as*by*. For process of reduction this value is known as*Interest*.*Decay* - Growth or reduction happens
some value. For process of growth this value is known as*to*. For process of reduction this value is known as*Value After Interest*.*Value After Decay*

**Initial quantity of entity present:**The principal quantity**P**.**Current quantity of entity present:**This may or may not influence the extent of growth or reduction depending on the type of growth or reduction.**Time period of growth or reduction:**Time taken for a single unit of growth or reduction. It is denoted by letter**T.**This may or may not influence the extent of growth or reduction depending on the type of growth or reduction.**Rate of growth or reduction:**Amount by which the quantity grows or reduces during time**T**. It is denoted by letter**R.****No. of Intervals of growth or reduction:**Total no. of intervals of time**T**for which the growth or reduction has taken place. It is denoted by letter**t.**

**Arithmetic growth and reduction:**In arithmetic growth or reduction, both the valueswhich and*by*which the entities grow or reduce*to***are independent of**the current existing quantity of the entity, the actual interval for growth or reduction**T**and the no. of intervals of growth or reduction**t**.**Exponential growth and reduction:**In exponential growth or reduction, both the valueswhich and*by*which the entities grow or reduce are in terms of some factor of the current existing quantity of the entity and also depend on the actual interval for growth or reduction*to***T**and the no. of intervals of growth or reduction**t**.

- An increase in time period
**T**by a**factor**the rate*increases***R**by**same factor**andthe number of intervals*decreases***t**by**same factor**. This means if the value of**T**becomes**T x 3**, then value of**R**will become**R x 3**and value of**t**shall become**t/3.**That is, when we multiply**T**by a factor, we must multiply**R**by the same factor and divide**t**by the same factor - A decrease in time period
**T**by a**factor**the rate*decreases***R**by**same factor**andthe number of intervals*increases***t**by**same factor**. This means if the value of**T**becomes**T/5**, then value of**R**will become**R/5**and value of**t**shall become**t x 5**. That is, when we divide**T**by a factor, we must divide**R**by the same factor and multiply**t**by the same factor.

In arithmetic growth or reduction, both the values ** by** which and

For example let's assume a numerical entity

The value of

For the process of arithmetic growth

For the process of arithmetic reduction

- The value of the entity can become zero after certain intervals of time in case of reduction
- The value of amount generated after any no. of intervals of time is independent of the time period
**T**. (This is because any change in**T**shall bring about mutually canceling changes in both**R**ant**t)**. This property is illustrated below.

For e.g. suppose**P**=10,**T=**4,**t=**2,**R=**5

**Initial value= P (=10)**

**Value after time t (=2)**x**T (=4) = 10 + 2**x**(5**x**10) = 110**

Now, suppose we increase the time period**T**by a factor of**2**, so**T**becomes**Tx2= 4 x 2=****8**

The corresponding new value of**R**will be**= R x 2 = 5 x 2 =10**

The corresponding new value of**t**will be**= t / 2 = 2 / 2 =1**

**So, value after time t (=1)**x**T (=8)=10 + 1**x**(10**x**10) = 110**

Now, instead of increasing, if we decrease the time period**T**by a factor of**2**,**T**becomes**T/2= 4 / 2=****2**

The corresponding new value of**R**will be**= R / 2 = 5 / 2 =2.5**

The corresponding new value of**t**will be**= t x 2 = 2 x 2 =4**

**So, value after time t (=4)**x**T (=2)=10 + 4**x**(2.5**x**10) = 110**

The formula **P + t **x** (R **x** P) ** is also known as the formula for amount generated after simple interest.
In this formula the meaning of symbols is the following.

**P= **Initial amount deposited in the bank

**R**= Rate of interest **generally** **provided on a yearly basis and in percentage** (therefore it needs to be divided by hundred whenever such is the case).

**T=**Time period for interest calculation. **Generally this is a single year**.

**t=**Total time in years for which interest is calculated**. **It is some multiple of **T**. It can be in fraction.

Whenever the value of **R** is not given for the time period **T** of interest calculation, or the value of **t** is not given as a multiple of period **T** of interest calculation, they
need to be multiplied by some factor so as to bring them all in same time units. In such scenarios the formula for amount after simple interest becomes

**P + m **x** t **x** (n **x** R **x** P)**

Here **m** and **n** are the factors by which **R** and **t** need to be multiplied so as to bring them in terms of **T**. The factors **m** and **n** are reciprocals of each other.

In exponential growth or reduction, both the values ** by** which and

For example let's assume a numerical entity

Instead of doubling, if the value of

The value of

The

**Initial value=P**

**Value after time T = P **x** F**

**Growth in value after time T = (P **x** F) - P = P **x** (F-1)**

**So,value after time T = P + P **x** (F-1) = P **x** (1 + (F-1)) = P **x** F = P'**

**Similarly value after time 2 **x** T = P'** x **F**

**Growth in value after time 2 **x** T = P' **x** F - P' = P' **x** (F-1)**

**So, value after time 2 x T =P'+ P' **x** (F-1) = P' **x** (1 + (F-1)) = P **x** (1 + (F-1)) **x** (1 + (F-1)) = P **x** (1 + (F-1)) ^{2} = P **x

The quantity

\(\Rightarrow\)

The rate

Also, in terms of

**Initial value=P**

**Value after time T = P** x **F**

**Reduction in value after time T = P - (P **x** F) = P **x** (1-F)**

**So, value after time T = P - P **x** (1-F) = P **x** (1 - (1-F)) = P **x** F = P'**

**Similarly value after time 2** x **T = P'** x **F**

**Reduction in value after time 2 **x** T = P' - P' **x** F = P' **x** (1-F)**

**So, value after time 2 x T =P' - P' **x** (1-F) = P' **x** (1 - (1-F)) = P **x** (1 - (1-F)) **x** (1 - (1-F)) = P **x** (1 - (1-F)) ^{2} = P **x

The quantity

\(\Rightarrow\)

The rate for the process of reduction can be any real value

Also, in terms of

- The value of the entity can never become zero after any intervals of time in case of reduction.
- The value of quantity after any no. of intervals of time is dependent on the time period
**T**. A**decrease**in time period of growth (or reduction)**always increases**the growth factor. This property is illustrated below.

For e.g. suppose**P**=10,**T=**4,**t=**2,**R=**5

**Value after time t (=2)**x**T (=4) = P**x**(1+R)**x^{t}= 10**(1+5)**^{2}= 360

Now, suppose we increase the time period**T**by a factor of**2**, so**T**becomes**T x 2= 4 x 2=****8**

The corresponding new value of**R**will be =**R**x**2 = 5**x**2 =10**

The corresponding new value of**t**will be**= t/2 = 2 / 2 = 1**

**So, value after time t (=1)**x**T (=8) = P x (1+R)**x^{t}=10**(1+5x2)**^{1}= 110

Now, suppose instead of increasing, we decrease the time period**T**by a factor of**2**, so**T**becomes**T/2= 4 / 2=****2**The corresponding new value of**R**will be**= R/2 = 5 / 2 =2.5**

The corresponding new value of**t**will be**= tx2 = 2 x 2 =4**

**So, value after time t (=4)**x**T (=2)= P x (1+R)**x^{t}= 10**(1+2.5)**^{4}= 15006.25

The above two examples clearly show that the value of the factor of exponential growth or reduction is actually dependent upon the value of Time Period**T. The value of the Growth / Reduction Factor increases with a decrease in Time Period, and decreases with an increase in Time Period.**

The formula **P **x** (1+ R) ^{t} **is also known as the formula for calculation of amount generated after compound interest. In this formula the meaning of symbols is the following

Whenever the value of

Here

If we keep on decreasing the time period **T** for a given rate **R** and time interval **t **such that the process of growth or reduction takes place every moment.then such kind of growth or
reduction is known as **continuous growth or reduction**. In such cases, actual factor by which exponential growth takes place is mathematically represented as

**Continuous Growth Factor = (1 + R/n) ^{nt}** (such that

This growth factor is generally represented by powers of an irrational constant number

where

Since,

If the value of

On binomial expansion of the RHS of the above formula we get

which is the exponential series

If the value of

On binomial expansion of the RHS of the above formula we get

which is, once again the exponential series

Based on above definitions of

On binomial expansion of the RHS of the above formula we get

which is the exponential series for