Numerical sequences

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: u_{n+1}=q\times u_{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(u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} be the geometric sequence with first term u₀=5 and reason q=-2. Calculate u₁,u₂ and u₃

The answer:

u_{1}=(-2)\times u_{0}=(-2)\times 5=(-10).\\ u_{2}=(-2)\times u_{1}=(-2)\times (-10)=20.\\ u_{3}=(-2)\times u_{2}=(-2)\times 20=(-40).

Explicit formula

Proposition

If u is a geometric sequence of reason q(q\neq 0) then, for all natural numbers n and p,u_{n}=u_{p}\times q^{n-p}. In particular, for any natural number n: u_{n}=u_{0}\times q^{n}

Example

a/ Let (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} 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 u_{n}=u_{0}\times q^{n}

u_{1}=3\times 2^{1}=3\times 2=6.\\ u_{7}=3\times 2^{7}=3\times 128=384.\\ 2/ u_{5}=3\times 2^{5}=3\times 32=96.

b/ Let (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} be a geometric sequence of reason 12 and u₃=-40. Calculate u₆.

The answer: for all natural numbers n and p, u_{n}=u_{p}\times q^{n-p}

So for n=6 and p=3,

u_{6}=u_{3}\times 12^{(9-3)}=-40\times 12^{6}=-40\times 2985984=-119439360.

Partial sum

Theorem

The n-th partial sum (S[1]) of an geometric sequence(u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} with u_{n}=u_{p}\times q^{n-p} (where u_{p} is the first item) is given by:

\begin{equation*} S_{n}=u_{p}\times (\frac{1-q^{n-p+1}}{1-q}) \end{equation*}

Example

Consider a geometric sequence with u₀=2 and q=\dfrac{-4}{3}, calculate the partial sum S=u₀+⋯+u₅

The answer: According to Theorem we obtain the fifth partial sum as follows:

\begin{eqnarray*} S_{5} &=&u_{0}\times (\frac{1-(\dfrac{-4}{3})^{5-0+1}}{1-(-\dfrac{4}{3})}) \\ &=&2\times (\frac{1-(\dfrac{-4}{3})^{6}}{\dfrac{7}{3}}) \\ &=&2\times \frac{3}{7}\times (1-\frac{4096}{729}) \\ &=&\frac{6}{7}\times (\frac{-3367}{729})=\frac{-20202}{5103} \\ &=&\frac{-6734}{1701} \end{eqnarray*}

Compound Interest

Compound interest [2] 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[3] the future value  after n interest periods is given by:

\begin{equation*} CA=P(1+\frac{r}{m})^{mT} \end{equation*}

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 :

CI=CA-P.
Example

1/ Suppose 1000€ is invested for seven years at 12% compounded quarterly.

Determine the future value (the compound amount)?

Solution

We have : \begin{equation*}
CA=P(1+\frac{r}{m})^{mT}
\end{equation*}

where :P=1000€

m= 4.

r=12%=0.12 interest calculated 4 times a year.

t=7 years.

\begin{eqnarray*} CA &=&1000(1+\frac{0.12}{4})^{4\times 7} \\ &=&1000(1+0.03)^{28} \\ &=&1000\times 1.03^{28} \\ &=&2287.92 \end{eqnarray*}
Example

2/ What is the nominal rate compounded monthly that will make 1,000€ become 2,000€ in five years?

Solution

We have : \begin{equation*}
CA=P(1+\frac{r}{m})^{mT}
\end{equation*}

where :CA=2000€

P=1000€

m= 12.

r=? interest calculated 12 times a year.

T=5 years.

\begin{eqnarray*} 2000 &=&1000(1+\frac{r}{12})^{12\times 5} \\ \frac{2000}{1000} &=&\frac{1000}{1000}(1+\frac{r}{12})^{60} \\ 2 &=&(1+\frac{r}{12})^{60} \\ 2^{\frac{1}{60}} &=&(1+\frac{r}{12}) \\ 1.0116 &=&(1+\frac{r}{12}) \\ \frac{r}{12} &=&1.0116-1 \\ r &=&0.0116\times 12 \\ r &=&0.1392=13.92\% \end{eqnarray*}

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:

  1. S: Partial sum

  2. Compound

    Something that is formed by combining two or more parts.

  3. CA: Compound amount

PreviousPreviousNextNext
HomepageHomepagePrintPrintCreated with Scenari (new window)