AnnotationsQuit

Arithmetic sequences

Arithmetic sequence of reason r

Definition

A sequence u is said to be arithmetic if there exists a real number r such that for any integer natural number n, u_{n+1}=u_{n}+r. The real number r is called the reason of the sequence u.

Note

In other words, a sequence is arithmetic if and only if each term (except the first) is obtained by adding a real number r, always the same, to the previous term.

Example

Let (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}be the arithmetic sequence with first term u₀=5 and reason r=4. Calculate u₁,u₂ and u₃.

The answer: We can find each term by adding the reason r to the previous term

u_{1}=u_{0}+r=5+4=9.\\ u_{2}=u_{1}+r=9+4=13.\\ u_{3}=u_{2}+r=13+4=17.

Proposition

An arithmetic sequence of reason r is increasing if and only if r>0 and decreasing if and only if r<0.

Explicit formula

If u is an arithmetic sequence of reason r, then for all natural numbers n and p u_{n}=u_{p}+(n-p)r,

In particular, for any natural number n: u_{n}=u_{0}+nr.

Example

a/ Let (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} be the arithmetic sequence with first term u₀=8 and reason r=2

1/ For a natural number n, give the expression of the sequence (u_{n}) as a function of n.

2/ Calculate u₁ and u₇.

3/ Calculate the term at rank 12.

The answer:1/ We know that for any natural number n: u_{n}=u_{0}+nr then u_{n}=8+2n

2/

2/ u_{1}=8+2\times 1=8+2=10.\\ u_{7}=8+2\times 7=8+14=22.\\ 3/ u_{12}=8+2\times 12=8+24=32.

b/ Let (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} be an arithmetic sequence of reason 8 and u₃=-40. Calculate u₉.

The answer: for all natural numbers n and p,u_{n}=u_{p}+(n-p)r

So for n=9 and p=3,u_{9}=u_{3}+(9-3)\times 8=-40+6\times 8=-40+48=8

Partial sums

Theorem

The n-th partial sum of an arithmetic sequence (u_{n})_{n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
} with u_{n}=u_{p}+(n-p)r(where u_{p} is the first item) is given by

\begin{equation*} S_{n}=\frac{(n-p+1)}{2}(u_{p}+u_{n}) \end{equation*}

Example

We determine the total car production within the first twelve months of production. To this end, we have to determine the partial sum of an arithmetic sequence S=u₁+⋯+u₁₂with u₁=750 and r=20. We obtain

\begin{eqnarray*} S_{12} &=&\frac{(12-1+1)}{2}(u_{1}+u_{12}) \\ &=&\frac{12}{2}(u_{1}+u1+(12-1)r) \\ &=&6(2u_{1}+11r) \\ &=&6(2\times 750+11\times 20) \\ &=&6\times (1500+220) \\ &=&6\times 1720 \\ &=&10320 \end{eqnarray*}

the total car production within the first year is equal to 10,320.

Simple interest

Simple interestis a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time. In simple interest, the principal amount is always the same.

In this section, we will introduce the concept of borrowing money and the simple interest that is derived from borrowing. We will also introduce terms such as principal, amount, rate of interest, and time period. Through these terms, we can calculate simple interest using the simple interest formula.

What is Simple Interest?

Simple interest is a method of interest that always applies to the original principal amount, with the same rate of interest for every time cycle. When we invest our money in any bank, the bank provides us interest on our amount. The interest applied by the banks is of many types and one of them is simple interest. Now, before going deeper into the concept of simple interest, let's first understand what is the meaning of a loan.

A loan is an amount that a person borrows from a bank or a financial authority to fulfil their needs. Loan examples include home loans, car loans, education loans, and personal loans. A loan amount is required to be returned by the person to the authorities on time with an extra amount, which is usually the interest you pay on the loan.

Simple Interest Formula

SI is calculated with the following formula: SI=\dfrac{P\times
R\times T}{100}, where P= Principal, R= Rate of Interest in % per annum, and T= Time, usually calculated as the number of years. The rate of interest is in percentage R% (and is to be written as (\dfrac{R}{100}), thus 100 in the formula).

Principal: The principal is the amount that was initially borrowed (loan) from the bank or invested. The principal is denoted by P.

Rate: Rate is the rate of interest at which the principal amount is given to someone for a certain time, the rate of interest can be 5%,10%, or 13%, etc. The rate of interest is denoted by R.

Time:  is the duration for which the principal amount is given to someone. Time is denoted by T.

The above formula can be further solved for any variable, P,R, or T. For example, by dividing both sides of the SI formulaSI=\dfrac{P\times
R\times T}{100} by R×T, we get P=\dfrac{100\times SI}{R\times \ T}. Similarly, we can solve for either R or T.

Simple interest formula:

\begin{eqnarray*} SI &=&\frac{P\times R\times T}{100} \\ P &=&\frac{SI\times 100}{R\times T} \\ R &=&\frac{SI\times 100}{P\times T} \\ T &=&\frac{SI\times 100}{R\times P} \end{eqnarray*}

Sometimes, the simple interest formula is written as just SI=PRT where R is the rate of interest as a decimal. i.e., if the rate of interest is 5% then R can be written as \dfrac{5}{100}=0.05.

Simple amount: When a person takes a loan from a bank, he/she has to return the principal borrowed plus the interest amount, and this total returned is called the Amount.

SA= Principal + Simple Interest

\begin{equation*} \begin{array}{ll} SA & =P+S.I \\ & =P+PRT \\ & =P(1+RT)% \end{array}% \end{equation*}
Example

Mohamed had borrowed 100.000DA from the bank and the rate of interest was 5%. What would the simple interest be if the amount is borrowed for 1 year? Similarly, calculate the simple interest if the amount is borrowed for 2 years, 3 years, and 10 years? Also, calculate the amount that has to be returned in each of these cases.

Solution

We have : Principal Amount = 100.000DA, Rate of Interest R=5%=\dfrac{5}{100}=0.05.

\begin{equation*} \begin{tabular}{|c|c|c|} \hline Duration & Simple interest & Amount \\ \hline 1 year & $SI=100000\times 0.05\times 1=5000$ & $A=100000+5000=105000$ \\ \hline 2 years & $SI=100000\times 0.05\times 2=10000$ & $A=100000+10000=110000$ \\ \hline 3 years & $SI=100000\times 0.05\times 3=15000$ & $A=100000+15000=115000$ \\ \hline 10 years & $SI=100000\times 0.05\times 10=50000$ & $A=100000+50000=150000$ \\ \hline \end{tabular}% \end{equation*}
Example

How much money was invested at 5% annual simple interest for 5 years to earn 500000DA?

Solution

Assume that principal value is P.

  • Rate of interest is, R=5%=0.05.

  • Time is, T=5years.

  • Amount is, A=500000.

Using the simple interest formula of amount,

\begin{eqnarray*} SA &=&P(1+RT) \\ 500000 &=&P(1+0.05\times 5) \\ 500000 &=&1.25P \\ P &=&500000\div 1.25 \\ &=&400000 \end{eqnarray*}

Then the invested money = 400000DA.