Geometric sequences
Geometric sequences of reason q
A sequence u is said to be geometric if there exists a real number q such that for any integer natural number n: . The real number q is called the reason of the sequence u.
Note :
In other words, we go from one term of the sequence to the next by always multiplying by q.
Example :
Let be the geometric sequence with first term u₀=5 and reason q=-2. Calculate u₁,u₂ and u₃
The answer:

Explicit formula
Proposition
If u is a geometric sequence of reason then, for all natural numbers n and p,
. In particular, for any natural number n:
Example :
a/ Let be the geometric sequence with first term u₀=3 and reason q=2.
1/ Calculate u₁ and u₇.
2/ Calculate the term at rank 5.
The answer:1/ We know that for any natural number

b/ Let be a geometric sequence of reason 12 and u₃=-40. Calculate u₆.
The answer: for all natural numbers n and p,
So for n=6 and p=3,

Partial sum
Theorem
The n-th partial sum (S) of an geometric sequence with
(where
is the first item) is given by:

Example :
Consider a geometric sequence with u₀=2 and , calculate the partial sum S=u₀+⋯+u₅
The answer: According to Theorem we obtain the fifth partial sum as follows:

Compound Interest
Compound interest is an interest calculated on the principal and the existing interest together over a given time period. The interest accumulated on a principal over a period of time is also added to the principal and becomes the new principal amount for the next time period. Again, the interest for the next time period is calculated on the accumulated principal value. Compound interest is the method of calculation of interest used for all financial and business transactions across the world. The power of compounding is that it is always greater than or equal to the other methods like simple interest.
What is compound interest ?
Compound interest computation is based on the principal which changes from time to time.
Interest that is earned is compounded /converted into principal & earns interest thereafter.
The principal increases from time to time.
Compound Interest Amount
The formula to calculate CA the future value after n interest periods is given by:

where :
CA : The total amount accumilated after T time periods
P : Principal amount (intial investment or loan amount)
m : Number of times that interest is compounded per year
r : nominal interest rate (per year)
T : Time in years, usually calculated as the number of years
The Compound Interest Formula :

Example :
1/ Suppose 1000€ is invested for seven years at 12% compounded quarterly.
Determine the future value (the compound amount)?
Solution
We have :
where :P=1000€
m= 4.
r=12%=0.12 interest calculated 4 times a year.
t=7 years.

Example :
2/ What is the nominal rate compounded monthly that will make 1,000€ become 2,000€ in five years?
Solution
We have :
where :CA=2000€
P=1000€
m= 12.
r=? interest calculated 12 times a year.
T=5 years.

Simple Interest vs Compound Interest
Simple interest and compound interest are two ways to calculate interest on a loan amount. It is believed that compound interest is more difficult to calculate than simple interest because of some basic differences in both. Let's understand the difference between simple interest and compound interest through the table given below:
