1. Introduction
About sequence:
From Latin word ‘sequens’,sequence is derived which means a sequence is a
collection of objects like numbers, letters and symbols, these can be listed in any
way possible irrespective of repetitions and order. Each sequence has a specific
length which is defined by the number of elements in the sequence.
Note: There is no end to the sequence and they are the basis for series, which are
important in differential equations and analysis.
Representation of sequence:
Each element has a rank (also known as index) and this rank defines the position of
the element. Following this the first element of each sequences either has rank of 0
or 1 depending on the sequence. A typical notation of a sequence would appear
like this:
{𝑎𝑛 } = {1, 4, 9, and 16}
The notation without braces being the ‘index’ or the counter for example ‘a2’
which simply designates the specific term in the sequence, in this case the second
term in that sequence. However, the notation with braces which then refers to the
entire sequence.
Recursion:
A recursion is another method to determine the elements of a sequence, slightly
altered index which uses the previous elements of the sequence to establish the
following elements. It is only applicable to sequences whose elements are related
to the previous elements. These sequences are denoted as recursive sequences
whose index is called a recursion.
2. Types of sequence:
1. Arithmetic sequence
Arithmetic sequences, also known as arithmetic progressions and the elements
of an arithmetic sequence are dependent upon the preceding elements of the
sequence. However, contrary to geometric sequences, the elements are
determined by adding and subtracting a certain number from the previous
elements of the sequence. All arithmetic sequences increase or decrease in a
linear manner which means constant .This number remains the same throughout
the sequence and is known as the common difference. The common difference
of an arithmetic sequence is determined by subtracting the first of two
sequential elements from the second, therefore the result is denoted ‘difference’
as‘d’.
For example:
The sequence (𝑎𝑛) = {23, 6, −11, −28, −45, −62……} the common difference
is – 17, because to determine the next element, one would need to subtract 17
from the previous element.
The notation of the index of the sequence above would be as follows: (𝑎𝑛) =
𝑎1 + (𝑛 − 1)
Properties:
The nature of an arithmetic sequence depends on the common difference.
If the common difference is positive, the elements will go on to positive
infinity.
If the common difference is negative, the elements will go on to negative
infinity.
3. Geometric sequence:
A geometric sequence, also known as a geometric progression, is a
sequence where each element is determined by multiplying or dividing the
previous elements by a fixed non-zero number. This number is known as the
common ratio. Apart from multiplications and divisions, powers can also be
the common ratio of a geometric sequence. These powers are represented as
follows: 𝑟 𝑑 𝑜𝑟 2𝑑. Typically, ‘r’ is any non-zero number.
For example:
The sequence (𝑎𝑛) = {1, 7, 49, 343 …} the common ratio is 7, because each
preceding element is always multiplied by 7 to determine the next element
of the sequence.
The sequence (𝑎𝑛 ) = {𝑎, 𝑎𝑟, … , 𝑎𝑟 ^𝑑 , 𝑎𝑟 ^𝑑+1 , 𝑎𝑟 ^𝑑+2 , … } is an
example of a geometric sequence where the common ratio of the sequence is
a power
Properties:
1 the index of all geometric sequences is based upon the preceding terms of the
sequence all geometric sequences follow a recursive relation.
2 To prove the validity of a geometric sequence, one would have to find a common
ratio that repeats itself across all elements.
3 If the index repeats itself on all elements the sequence becomes infinite and
grows exponentially large, only if the common ratio is 1 and doesn’t consist of a
division.
4. Series is the sum of a list of numbers that are generating according to some pattern
or rule.
For example, '1+3+5+7+9' is a mathematical series - the sum of the first five odd
numbers.
A series is a description of the operation of adding infinitely many quantities, one
after the other, to a given starting quantity.[1]
The study of series is a major part
of calculus and its generalization, mathematical analysis. Series are used in most
areas of mathematics, even for studying finite structures (such as in combinatory)
through generating functions. In addition to their ubiquity in mathematics, infinite
series are also widely used in other quantitative disciplines such
as physics, computer science, statistics and finance.
General example of series in real life:
Suppose that your roommate baked an apple pie and left it out on the counter while
she went off to work. Being very fair, you decide to only eat ½ the pie while she's
gone; however, an hour later, you're still hungry, so you eat ½ of what is now left,
or ¼ of the total pie.
Unfortunately for your roommate, her pie is delicious, so you find yourself every
hour again eating half of what's remaining.
As a fraction of the total pie, your pie eating during the eight hours she is at work
looks like this:
½, ¼, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256
Those last slivers of pie were pretty tiny, but you managed.
The list of fractions of pie is a sequence - it's simply a list with commas between
each number.
If you want to know how much of the pie you ate altogether, then you create a
series (a sum), that looks like this:
½ + ¼ + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256
By the time your roommate returns, you have polished off 255/256 of the pie.
5. 1. in Finance
a. Investment
For example if I invest Rs600 in a commodity at an 8% interest rate, I will be able
to calculate the investment made each year. This works by being able to calculate
8% percent onto the previous investment. For example, if I start out with an
investment of Rs600 the investment will continue like this: {Rs600, Rs648,
Rs699.84, Rs755.83, Rs816.29, Rs881.60, Rs952.12, and Rs1028.29}.3.
Calculating the Price of an Object
Depreciation
Depreciation is the constant loss of value of an object over a set period of time.
This means that the phenomenon is the same as investing, the only difference
being the fact that an amount is always subtracted from the total compared to
added.
Suppose you own a mobile that you purchased for some amount. If the price of the
mobile depreciates (decreases) by same percentage every year, you can find out the
value of the mobile after n years, using the geometric sequence.
For example, a mobile phone that is bought for Rs400 depreciates at a rate of 10%
for the first year and every year the rate drops by 1%.The elements of the sequence
would be ordered as follows: { Rs360, Rs327.6, Rs301.39, Rs280.29, Rs263.48}.
2. Exponential growth
The expansion rate of something can easily be predicted or foreseen, after having
applied this method. This can be used to predict population increase and the spread
of disease among others.
6. Examples: A deadly disease is being spread and scientists need to estimate the risk
and how many people could be affected by the disease. If an infected patient gives
the disease to 5 people every week and these people, then in turn also pass the
disease on to five other people the next week each. How many people would be
infected within a month?
Here, n represents the number of weeks and ‘i’ represents the previous element of
the sequence.
{1, 6, 31, 156, 781} [People infected each week]
(𝑎𝑛 ) = 5*𝑛−1 + 𝑎𝑖−1
After five weeks the number of people infected would be 5*7 + 19531 = 97656
Next example:
(b) What will be the total number of insects in the five generations?
By using the formula for the sum of the first n terms of a geometric sequence.
The total population for the five generations will be about 1319 insects.
7. 3. Movement
The action is carried out across many sports such as basketball, football and
many others.
For example: When bouncing a ball, it bounces at a certain rate and then
logically loses this energy over a certain time period. This process has been
studied and the conclusion was that the ball loses its energy at a same rate,
with the same fraction of energy being deducted every time the ball bounces.
And during a basketball game the players factor into this without even
thinking about it. The same applies if you are playing football with friends
and you take a free kick, you would need the factor in how much the ball
drops every second, which is something we all do. Everyone uses
sequencing to predict the trajectory of objects.
4. DNA Sequencing
DNA sequencing is used to determine the nucleic acid sequence and used in
the field of science. This information is vital importance for a researcher in
understanding the type of genetic information that is contained within our
DNA. It specifies the order in which the constituents find themselves in,
which could lead to the premature detection of certain genetic illness and
may even foresee and prevent illness occurring in the first place.
5. In Making Pyramid-like Structures
A sequence helps you design pyramid-like structures where the building
components change in a constant manner.
Example: A child building a tower with blocks uses 15 for the bottom row. Each
row has 2 fewer blocks than the previous row. Suppose that there are 8 rows in the
tower.
(a) How many blocks are used for the top row?
The number of blocks in each row forms an arithmetic sequence with a1 = 15
and d = −2. Find an for n = 8 by using the formula an = a1 + (n − 1) d.
a8 = 15 + (8 − 1)*(−2) a8 = 1.
There is just one block in the top row.
8. (b)What is the total number of blocks in the tower?
Here we must find the sum of the terms of the arithmetic sequence formed
with a1 = 15, n = 8, and a8 = 1 by using the formula
Sn = n/2(a1 + an).
S8 = 8/2(15 + 1)
S8 = 4(16) S8 = 64.
There are 64 blocks in the tower.
6. Compound Interest as Geometric Sequence
If you put your money in a bank and the bank provides you a fix annual rate of
interest, then you can calculate the amount you will have in your account after
certain years by using the concept of geometric progression.
Example: If Linda deposits $1300 in a bank at 7% interest compounded annually,
how much will be in the bank 17 years later?
Using the compound interest formula, A = P(1 + r/k)*kt
with P = principle, t = time in years, r = annual rate, and k = number of periods per
year.
A = 1300(1 + 0.07/1)*1• 17
A = 1300(1 + 0.07)17
A = 4106.46.
The account will contain $4,106.46.
7. Calculating simple interest
Example: Find the accumulated value of $15,000 at 5% per year for 18 years using
simple interest.
Using the simple interest formula, I = P*r*t with I = total interest, P = principle, r
= annual rate, and t = time in years.
I = 15,000(0.05)*(18) I =13,500.
A total of $13,500 in interest will be earned.
Hence, the accumulated value in the account will be 13500 + 15000 = $28,500.
9. Some other useful examples
Suppose you go to work for a company that pays one penny on the first day, 2
cents on the second day, 4 cents on the third day and so on. If the daily wage keeps
doubling, what will you total income be for working 31 days?
This problem is geometric because the problem says that the salary from the
previous day is doubled, or multiplied by 2. When we are multiplying by the same
number each time, this is a geometric sequence. But what do we need to do with
this geometric sequence?
The problem wants to know TOTAL income after 31 days. When dealing with
total amounts, like in the previous example, we need to add the terms in a
sequence. In this case, since we will be adding terms in a geometric sequence, we
will be finding a geometric series. So we need the formula for a geometric series
Similarly,
Thus, he earns almost 21.5 million dollars in a month.
10. Conclusion:
In conclusion, we came to know about the concept of sequence and series and their
real life examples in day to day life. Alternatively, a sequence is defined as an
arrangement of numbers in a particular order.
An example of sequence: 2, 4, 6, 8, …
On the other hand, a series is defined as the sum of the elements of a sequence.
An example of a series: 2 + 4 + 6 + 8 + …
Finite Sequences: A finite sequence is a sequence that contains the last term such
as a1, a2, a3, a4, a5, a6……an. On the other hand, an infinite sequence is never-ending
i.e. a1, a2, a3, a4, a5, a6……an…..
Finite Series: In a finite series, a finite number of terms are written like a1 + a2 +
a3 + a4 + a5 + a6 + ……an. In case of an infinite series, the number of elements are
not finite i.e. a1 + a2 + a3 + a4 + a5 + a6 + ……an +……….
Similarly, applications of sequence and series in real life has unlimited examples
which are explained above. This real life examples showed us that mathematics
may be a subject of interest after knowing its real life applications rather than
theory.
World is nothing without the use of mathematics and it is not only used in one
particular sector but also in many other sectors i.e. multiple areas. Furthermore, it
can also be used to calculate the no of tourists that will be visiting our country,
similarly in the field of economy tooo.so it is not only used in business but also in
many different sectors.
11. ACKNOWLEDGEMENT
we are over helmed in all humbleness and gratefulness to acknowledge our depth
to all those who have helped us to put these ideas, well above the level of
simplicity and into something concrete.
we would like to express our special thanks of gratitude to our teacher who gave
us the golden opportunity to do this wonderful project on the topic "Application of
sequence and series on daily life" , which also helped us in doing a lot of Research
and we came to know about so many new things. We are really thankful to them.
Any attempt at any level can 't be satifactorily completed without the support and
guidance of our friends.
we would like to thank our friends who helped us a lot in gathering different
information, collecting data and guiding us from time to time in making this
project , despite of their busy schedules ,they gave us different ideas in making this
project unique.
Thanking you,
12. Project Work
Kathmandu Model Secondary School
Balkumari, Lalitpur
Project Work 1
Project Work Submitted By
Name: Faculty:
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Submitted To
Department of english
Kathmandu Model Secondary School
Balkumari, Lalitpur
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Table of content
14. BIBILOGRAPHY
1.Class 11 basic maths.
2.https://prezi.com/awocnux8p_pn/sequence-and-series-application
3.https://www.quora.com/How-can-you-apply-series-and-sequences in
real life
