A TEAMPAPER ON

SIMPLE INTEREST AND ARITHMETIC SEQUECNCECourse name: Business Mathematics

Course code: BUS 1205

Submitted to……..

M.A. KALAM AZADAssistant Professor

Submitted by SHOWKAT AHMED

Batch-BBA03 Section-A

Army Institute of Business Administration

Savar, Dhaka

Declaration I hereby declare that this paper/Dissertation/report has been written by me and not been previously submitted to any other University/Institute/Journal/Organization.

This work have presented does not have breach any copyright.

(……………………………)

SHOWKAT AHMED

ID-B3160B034

Section-A

BBA-03

AcknowledgmentI would first like to thank my Term paper advisor M.A. KALAM AZAD of the Assistant Professor at Army Institute of Business Administration. The door to sir office was always open whenever I ran into a trouble spot or had a question about my research or writing. He consistently allowed this paper to be my own work, but steered me in the right the direction whenever he thought I needed it.

AbstractInterest means the extra money added with principal as profit & Arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms. Interest is denoted by “I”. And the formula is “I = Prt” where P = Principal, R = Rate of Interest, T= Net Time. If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The fixed amount is called the common difference, d, referring to the fact that the difference between two successive terms yields the constant value that was added.

There are two kind of methodology in term paper. Frist in primary data and second in secondary data. Secondary data is when we collect information in different book and collect in internet. I collect much information in book and internet. So this is secondary data methodology.

Simple interest problems can involve lending or borrowing. In both cases the same formulas are used. Whenever money is borrowed, the total amount to be paid back equals the principal borrowed plus the interest charge. E.G. Given: principal: 'P' = $1500, interest rate: 'R' = 12% = 0.12, repayment time: 'T' = 2 years

Part 1: Find the amount of interest paid. Part 2: Find the total amount to be paid back.

Interest: 'I' = PRT Total repayments = principal + interest

= 1500 × 0.12 × 2 = $1500 + $360

=360 = $1860

Formula of Arithmetic Sequence is

Nth term = a1+(n-1)d Summation = n/2 {2a+(n-1)d}

Finally, we can say that it’s an essential part of business because; we can see its practical application in business sector. Its an part & parcel in business mathematics which is an essential course for business student. Simple Interest is the necessitate part of Banking Sector. So, that’s why we need to learn both simple interest & arithmetic sequence.

Table of Content

SL. No Description Page No01 Cover Page I02 Declaration Ii03 Acknowledgement Iii04 Abstract Iv05 Introduction 0106 Methodology 0407 Research Work 0508 Conclusion 15

CHAPTER-01

INTRODUCTION

Simple Interest

When someone lends money to someone else, the borrower usually pays a fee to the lender. This fee is called 'interest'. 'Simple' interest or 'flat rate' interest. The amount of simple interest paid each year is a fixed percentage of the amount borrowed or lent at the start.

Examples:

Example 1:

A credit union has issued a 3-year loan of $5,000. Simple interest is charged at a rate of 10 per cent per year. The principle plus interest is to be repaid at the end of the third year. Compute the interest for three year period. What amount will be repaid at the end of the third year?

Solution:

I = Pin

I = ($5,000)(0.10)(3)

= $1,500

The amount to be repaid is the principle plus the accumulated interest, that is:

$5,000 + $1,500

$6,500

Example 2:

A person lends $10,000 to a corporation by purchasing a bond from the corporation. Simple interest is computed quarterly at a rate of 3 per cent per quarter, and a check for the interest is mailed each quarter to all bondholders. The bonds expire at the end of 5 years, and the final check includes the original principle plus interest earned during the last quarter. Compute the interest earned each quarter and the total interest which will be earned over 5-year life of the bonds.

Solution:

In this problem P = $10,000, i = 0.03 per quarter, and the period of the loan is 5 years. Since the time period for i is a quarter (of a year), we must consider 5 years as 20 quarters. And since we

are interested in the amount of interest earned over one quarter, we must let n = 1. Therefore quarterly interest equals:

I = Pin

I = ($10,000)(0.03)(1)

= $300

To compute the total interest over the five year period, we multiply the per-quarter interest of $300 by the number of quarters, 20, to obtain

Total interest = $300 × 200

= $6000

Arithmetic Sequence

Arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms.

If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same).

The fixed amount is called the common difference, d, referring to the fact that the difference between two successive terms yields the constant value that was added. To find the common difference, subtract the first term from the second term.

Notice the linear nature of the scatter plot of the terms of an arithmetic sequence. The domain consists of the counting numbers 1, 2, 3, 4,... and the range consists of the terms of the sequence. While the x value increases by a constant value of one, the y value increases by a constant value of 3 (for this graph).

Example-

Arithmetic Sequence Common Difference, d

1, 4, 7, 10, 13, 16, ... d = 3Add 3 to each term to arrive at the next term, or...the difference a2 - a1 is 3.

15, 10, 5, 0, -5, -10, ... d = -5Add -5 to each term to arrive at the next term,or...the difference  a2 - a1 is -5.

Add -1/2 to each term to arrive at the next term, or....the difference a2 - a1 is -1/2.

CHAPTER-02

METHODLOGYThere are two kind of methodology in term paper. First in primary data and second in secondary data. Secondary data is when we collect information in different book and collect in internet. When I make term paper about “SIMPLE INTEREST AND ARITHMETIC SEQUENCE” I collect much information in book and internet. So this is secondary data methodology.

CHAPTER-03

RESEARCH WORK

3.1. The simple interest formula is as follows:

Interest = Principal × Rate × Time

Where:

'Interest' is the total amount of interest paid,

'Principal' is the amount lent or borrowed,

'Rate' is the percentage of the principal charged as interest each year. The rate is expressed as a decimal fraction, so percentages must be divided by 100. For example, if the rate is 15%, then use 15/100 or 0.15 in the formula.

'Time' is the time in years of the loan.

3.1.1 The simple interest formula is often abbreviated in this form:

I = P R T

Three other variations of this formula are used to find P, R and T:

Simple interest problems can involve lending or borrowing. In both cases the same formulas are used.

Whenever money is borrowed, the total amount to be paid back equals the principal borrowed plus the interest charge:

Total repayments = (principal + interest)

Usually the money is paid back in regular instalments, either monthly or weekly. To calculate the regular payment amount, you divide the total amount to be repaid by the number of months ( or weeks ) of the loan.

To convert the loan period, 'T', from years to months, you multiply it by 12, since there are 12 months in a year. Or, to convert 'T' to weeks, you multiply by 52, because there are 52 weeks in a year.

3.1.2 The example problem below shows you how to use these formulas:

Example:

A student purchases a computer by obtaining a simple interest loan. The computer costs $1500, and the interest rate on the loan is 12%. If the loan is to be paid back in weekly instalments over 2 years, calculate:

1. The amount of interest paid over the 2 years,

2. The total amount to be paid back,

3. The weekly payment amount.

Given: principal: 'P' = $1500, interest rate: 'R' = 12% = 0.12, repayment time: 'T' = 2 years

Part 1: Find the amount of interest paid.

Interest: 'I' = PRT

= 1500 × 0.12 × 2

= $360

Part 2: Find the total amount to be paid back.

Total repayments = principal + interest

= $1500 + $360

= $1860

Part 3: Calculate the weekly payment amount

Total repayments

Weekly payment amount = ---------------------------------------

Loan period, T, in weeks

$1860

= -------------------

2 × 52

= $17.88 per week

3.1.3 Identification of Simple

Simple interest can be identified by a simple math formula. The principal amount of the loan is multiplied by the rate of interest paid per year. That total is then multiplied by the number of years of the loan. The result is the simple interest. Simple interest is usually stated as an annualized percentage rate. This is the amount of interest that would be paid over 1 year, even if the term is shorter or longer.

3.1.4 Features of Simple interest

Simple interest does not consider the interest charged on interest. When a bank pays interest on a savings account, the amount in that account increases. If that interest is left in the account, there will be a larger amount for the bank to pay interest on the next time. This is called compound interest. The amount the account actually accumulates over the course of a year after interest is compounded several times is called the annual percentage yield. Simple interest is the percentage calculated each time, not the amount actually accrued.

3.1.5 Considerations of Simple interest

There are several things to consider when comparing various bank accounts or loans paying simple interest. The term is just as important as the interest rate. A savings account that pays 2 percent every 6 months is not the same as one that pays 4 percent every year. This is because the 6-month account would compound once during that year. It is also important to consider whether the interest is paid separately from the account or added into the account. If the interest is kept separate, it is simple interest. If the interest is added to the account, it is compound interest.

3.1.6 Function of Simple interest

Simple interest is sometimes used for very short-term loans. For example, simple interest can be charged for a period of 30 days when the short-term loan is promised to be repaid. The repayment amount is the principal loan plus the simple interest. Since the period of time is so short, the interest is not compounded.

3.1.6 Benefits of Simple interest

Some mortgages are offered with simple interest. They are identical to traditional mortgages except for the frequency with which the interest is calculated. Simple-interest mortgages calculate interest every day, while traditional mortgages do it once per month. Simple-interest mortgages benefit people who want to pay off their mortgages early. When mortgage payments are made more frequently and for greater amounts, the simple interest is calculated on a smaller mortgage balance. Simple-interest mortgages are bad for people who wait until the last minute to make their mortgage payment and need all month to gather the money. Since interest is calculated every day, they end up paying more in interest.

3.2.1 Formula of Arithmetic sequence

In General we could write an arithmetic sequence like this:

{a, a+d, a+2d, a+3d, ... }

Where:

Formulas used with arithmetic sequences and arithmetic series:

To find any term of an arithmetic sequence:

where a1 is the first term of the sequence,

d is the common difference, n is the

To find the sum of a certain number of terms of an arithmetic sequence:

where Sn is the sum of n terms (nth partial

number of the term to find.Note:  a1 is often simply referred to as a.

sum), a1 is the first term,  an is the nth term.

Examples:

Question Answer

1.  Find the common difference for this arithmetic sequence                          5, 9, 13, 17 ...

1.  The common difference, d, can be found by subtracting the first term from the second term, which in this problem yields 4.  Checking shows that 4 is the difference between all of the entries.

2.  Find the common difference for the arithmetic sequence whose formula is                          an = 6n + 3

2. The formula indicates that 6 is the value being added (with increasing multiples) as the terms increase.  A listing of the terms will also show what is happening in the sequence (start with n = 1).                           9, 15, 21, 27, 33, ...The list shows the common difference to be 6.

3.  Find the 10th term of the sequence                          3, 5, 7, 9, ...

3. n = 10;  a1 = 3, d = 2

 The tenth term is 21.

4.  Find a7 for an arithmetic sequence where                  a1 = 3x and d = -x.

4.  n = 7;  a1 = 3x, d = -x

5.  Find  t15 for an arithmetic sequence where          t3 = -4 + 5i  and  t6 = -13 + 11i

5.  Notice the change of labeling from a to t.  The letter used in labeling is of no importance.  Get a visual image of this problem 

Using the third term as the "first" term, find the common difference from these known terms.

Now, from t3 to t15 is 13 terms.t15 = -4 + 5i + (13-1)(-3 +2i) = -4 + 5i -36 +24i     = -40 + 29i

6.  Find a formula for the sequence                          1, 3, 5, 7, ...

6.  A formula will relate the subscript number of each term to the actual value of the term.

              Substituting n = 1, gives 1.Substituting n = 2, gives 3, and so on.

7.  Find the 25th term of the sequence                       -7, -4, -1, 2, ...

7.  n = 25;  a1 = -7, d = 3

8.  Find the sum of the first 12 positive even      integers.

8.  The word "sum" indicates the need for the sum formula.positive even integers:  2, 4, 6, 8, ...     n = 12;  a1 = 2, d = 2We are missing a12, for the sum formula, so we use the "any term" formula to find it.

Now, let's find the sum:

9.  Insert 3 arithmetic means between 7 and 23.

Note:  An arithmetic mean is the term between any two

9.  While there are several solution methods, we will use our arithmetic sequence formulas.Draw a picture to better understand the situation.                 7, ____, ____, ____, 23This set of terms will be an arithmetic sequence.

terms of an arithmetic sequence.  It is simply the average (mean) of the given terms.

We know the first term, a1,  the last term, an, but not the common difference, d.  This question makes NO mention of "sum", so avoid that formula.Find the common difference:

Now, insert the terms using d.7, 11, 15, 19, 23

10.  Find the number of terms in the sequence                7, 10, 13,..., 55.

10.   a1 = 7, an = 55,  d = 3.  We need to find n.This question makes NO mention of "sum", so avoid that formula.

When solving for n, be sure your answer is a positive integer.  There is no such thing as a fractional number of terms in a sequence!

11.  A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern.  If the theater has 20 rows of seats, how many seats are in the theater?

11. The seating pattern is forming an arithmetic sequence.    60, 68, 76, ...We wish to find "the sum" of all of the seats.n = 20,  a1 = 60,  d = 8 and we need a20 for the sum.

Now, use the sum formula:

There are 2720 seats.

The formula for the first n terms of an arithmetic sequence, starting with n = 1, is:

The sum is, in effect, n times the "average" of the first and last terms. This sum of the first n terms is called "the n-th partial sum".

3.2.2 Find the 35th partial sum of an = (1/2)n + 1

The 35th partial sum of this sequence is the sum of the first thirty-five terms. The first few terms of the sequence are:

a1 = (1/2)(1) + 1 = 3/2 a2 = (1/2)(2) + 1 = 2 a3 = (1/2)(3) + 1 = 5/2

The terms have a common difference d = 1/2, so this is indeed an arithmetic sequence. The last term in the partial sum will be a35 = a1 + (35 – 1)(d) = 3/2 + (34)(1/2) = 37/2. Then, plugging into the formula, the 35th partial sum is:

(n/2)(a1 + an) = (35/2)(3/2 + 37/2) = (35/2)(40/2) = 350

3.2.3 Find the value of the following summation:

From the formula ("2n – 5") for the n-th term, I can see that each term will be two units larger than the previous term. (Plug in values for n if you're not sure about this.) So this is indeed an arithmetical sum. But this summation starts at n = 15, not at n = 1, and the summation formula applies to sums starting at n = 1. So how can I work with this summation?

The quickest way to find the value of this sum is to find the 14th and 47th partial sums, and then subtract the 14th from the 47th. By doing this subtraction, I'll be left with the value of the sum of the 15th through 47th terms. The first term is a1 = 2(1) – 5 = –3. The other necessary terms are a14 = 2(14) – 5 = 23 and a47 = 2(47) – 5 = 89.

Subtracting, I get:

By the way, another notation for the summation of the first fourteen terms is "S14", so the subtraction could also be expressed as "S47 – S14".

Formatting note: Since this was just a summation, it's safe to assume that "2n – 5" is the expression being summed. However (and especially if you're dealing with something more complex), sometimes grouping symbols may be necessary to make the meaning clear:

3.2.4 Find the value of n for which the following equation is true:

I know that the first term is a1 = 0.25(1) + 2 = 2.25. I can see from the formula that each term will be 0.25 units bigger than the previous term, so this is an arithmetical series. Then the summation formula for arithmetical series gives me:

(n/2)(2.25 + [0.25n + 2]) = 21 n(2.25 + 0.25n + 2) = 42 n(0.25n + 4.25) = 42 0.25n2 + 4.25n – 42 = 0 n2 + 17n – 168 = 0 (n + 24)(n – 7) = 0

Solving the quadratic, I get that n = –24 (which won't work in this context) or n = 7.

n = 7

You could do the above exercise by adding terms until you get to the required total of "21". But your instructor could easily give you a summation that requires, say, eighty-six terms before you get the right total. So make sure you can do the computations from the formula.

3.2.5 Find the sum of 1 + 5 + 9 + ... + 49 + 53.

Checking the terms, I can see that this is indeed an arithmetic series: 5 – 1 = 4, 9 – 5 = 4, 53 – 49 = 4. I've got the first and last terms, but how many terms are there in total?I have the n-th term formula, "an = a1 + (n – 1)d", and I have a1 = 1 and d = 4. Plugging these into the formula, I can figure out how many terms there are:

an = a1 + (n – 1)d 53 = 1 + (n – 1)(4) 53 = 1 + 4n – 4 53 = 4n – 3 56 = 4n 14 = n

So there are 14 terms in this series. Now I have all the information I need:

1 + 5 + 9 + ... + 49 + 53 = (14/2)(1 + 53) = (7)(54) = 378

ConclusionFrom our above discussion we can say that it’s an essential part of business because, we can see its practical application in business sector. We can also find it in business mathematics which is an essential course for business student as well as business graduate.

We can’t think of a bank without interest. Because their business is closely connected with interest. The profit of their business is the difference between interest taken against loan & given against deposit. So simple interest is a part and parcel of a bank.

Simple Interest Formula & its application

A new firm loses $2000 in the first month. But it’s profit increases by $400 in each succeeding month for the next year. What is its profit in the 12th month?

Ans.

Given,

A1 = $2000 profit = A1 + (n-1)d

d = $400 = -2000+(12-1)400

n = 12 = $2400

Arithmetic Sequence Formula & its application

That’s why we need to learn both simple interest & arithmetic sequence.

Referencebook- business math

example for notes of the class

https://www.informationvine.com/index?am=broad&q=formula%20arithmetic%20sequence%20sum&an=google_s&askid=66fceead-3c77-4c3a-80e1-be73d9d13bac-0-iv_gsb&kv=default&gc=0&dqi&qsrc=999&ad=semD&o=33802&l&af&_=1

http://www.purplemath.com/modules/series4.htm

https://en.wikipedia.org/wiki/Arithmetic_progression

http://www.loanboss.com/topic/advantages-simple-interest