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Sequences and series

Patterns, Sequences, Arithmetic & geometric progression
by

Tejaswi Shitole

on 2 February 2014

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Transcript of Sequences and series

Patterns, Sequences, A.P. & G.P.
Contents
What are patterns, sequences and series?
What are arithmetic progressions?
General term of an A.P.
Sum of n terms of an A.P.
Examples from daily life situations
What are geometric progressions?
General terms of a G.P.
Examples of G.P. from real life situations
A few more examples
Videos
Sources
What are patterns, sequences and series?
A pattern is a type of theme of recurring events or objects, sometimes referred to as elements of a set of objects.

- It is the arrangement of numbers in a definite order according to some rule.
For example, take the numbers
1, 3, 5, 7, 9, . . . .
Here, we seem to have a rule. We have a sequence of odd numbers
A series is a sum of the terms in a sequence. If there are n terms in the sequence and we evaluate the sum then we often write Sn for the result, so that
Sn = u1 + u2 + u3 + . . . + un .
A geometric progression, or GP, is a sequence where each new term after the first is obtained by multiplying the preceding term by a constant r, called the common ratio.
If the first term of the sequence is a then the geometric progression is
a, ar, a x rsquare, a x r cube, . . .
where the nth term is a x r to the power of n-1.
Examples:
1) 2, 6, 18, 54, . . . .
Here, each term in the sequence is 3 times the previous term.
2) 1, −2, 4, −8, . . . ,
each term is −2 times the previous term.
A few more examples
Sources
http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-apgp-2009-1.pdf
Google Images
purplemath.com
R.D. Sharma
http://www.ilovemaths.com/3apsum.asp

Patterns in Nature
Patterns in Architecture
Which term of the A.P. 3,8,13 …is 78?

Solution: Here an = a + (n – 1) d = 78

a= 3, d = 8- 3 = 5

Therefore,

3 + (n -1) (5) = 78

(n-1) * 5 = 78 – 3 = 75

n – 1 = 75/5 = 15

n = 15 + 1 = 16

Hence 16th term (sixteenth term) is 78.
Find the tenth term and the n-th term of the following sequence:
1/2, 1, 2, 4, 8,...

The differences don't match: 2 – 1 = 1, but 4 – 2 = 2. So this isn't an arithmetic sequence. On the other hand, the ratios are the same: 2 ÷ 1 = 2, 4 ÷ 2 = 2, 8 ÷ 4 = 2. So this is a geometric sequence with common ratio r = 2 and a = 1/2. To find the tenth and n-th terms, I can just plug into the formula an = ar to the power (n – 1):

an = (1/2) 2 to the power of n–1
a10 = (1/2) 2 to the power of 10–1 = (1/2) 2 to the power of 9 = (1/2)(512) = 256
Find the n-th term and the first three terms of the arithmetic sequence having a6 = 5 and d = 3/2.

The n-th term of an arithmetic sequence is of the form an = a + (n – 1)d. In this case, that formula gives me a6 = a + (6 – 1)(3/2) = 5. Solving this formula for the value of the first term of the sequence, I get a = –5/2.
Then:
a1 = –5/2, a2 = –5/2 + 3/2 = –1, a3 = –1 + 3/2 = 1/2,
and an = –5/2 + (n – 1)(3/2)
Find the n-th term and the first three terms of the arithmetic sequence having a4 = 93 and a8 = 65.

Since a4 and a8 are four places apart, then I know from the definition of an arithmetic sequence that a8 = a4 + 4d. Using this, I can then solve for the common difference d:
65 = 93 + 4d
–28 = 4d
–7 = d
Also, I know that a4 = a + (4 – 1)d, so, using the value I just found for d, I can find the value of the first term a:

93 = a + 3(–7)
93 + 21 = a
114 = a

Once I have the value of the first term and the value of the common difference, I can plug-n-chug to find the values of the first three terms and the general form of the n-th term:

a1 = 114, a2 = 114 – 7 = 107, a3 = 107 – 7 = 100

an = 114 + (n – 1)(–7)
Find the n-th and the 26th term of the geometric sequence with a5 = 5/4 and a12 = 160.
These two terms are 12 – 5 = 7 places apart, so, from the definition of a geometric sequence, I know that a12 = ( a5 )( r7 ). I can use this to solve for the value of the common ratio r:

160 = (5/4)(r7)
128 = r7
2 = r

Since a5 = ar4, then I can solve for the value of the first term a:

5/4 = a(24) = 16a
5/64 = a

Once I have the value of the first term and the value of the common ratio, I can plug each into the formulas, and find my answers:

an = (5/64)2(n – 1)

a26 = (5/64)(225) = 2 621 440
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:

S14=n/2(a+an)

1 + 5 + 9 + ... + 49 + 53 = (14/2)(1 + 53) = (7)(54) = 378
What is a sequence?
- The various terms occurring in a sequence are called its terms.
- The terms are usually denoted by a1, a2, a3,...etc.
- The number at first place is called its first term of the sequence and is denoted by a1; the number at second place is denoted by a2 and so on.
- In general, the number at nth place is called the nth term of the sequence and is denoted by an.
- The nth term is also called the general term of the sequence.
Example:
Q1) Write the first three terms in each of the sequence defined by the following:
an=3n+2
Putting n=1,2,3 we get,
a1=3x1+2=3+2=5
a2=3x2+2=6+2=8
a3=3x3+2=9+2=11

Thus the required first three terms of the sequence defined by an=3n+2 are 5,8 ad 11.


Fibonacci's Sequence
The Fibonacci Sequence is the series of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...
The next number is found by adding up the two numbers before it.


The Fibonacci Sequence can be written as a "Rule"
First, the terms are numbered from 0 onwards like this:

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
xn= 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 ...
So term number 6 is called x6 (which equals 8).

Example: the 8th term is
the 7th term plus the 6th term: x8 = x7 + x6

So we can write the rule:

The Rule is xn = xn-1 + xn-2
where:
xn is term number "n"
xn-1 is the previous term (n-1)
xn-2 is the term before that (n-2)

Example: term 9 would be calculated like this:
x9 = x9-1 + x9-2 = x8 + x7 = 21 + 13 = 34

Types of sequences:
If the sequence goes on forever it is called an infinite sequence,
otherwise it is a finite sequence

Examples:
{1, 2, 3, 4 ,...} is a very simple sequence (and it is an infinite sequence)
{20, 25, 30, 35, ...} is also an infinite sequence
{1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a finite sequence)
{4, 3, 2, 1} is 4 to 1 backwards
{1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles
{a, b, c, d, e} is the sequence of the first 5 letters alphabetically
{f, r, e, d} is the sequence of letters in the name "fred"
{0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s (yes they are in order, it is an alternating order in this case)
Special Types of Sequences:
Triangular Numbers:

1, 3, 6, 10, 15, 21, 28, 36, 45, ...
The Triangular Number Sequence is generated from a pattern of dots which form a triangle.

By adding another row of dots and counting all the dots we can find the next number of the sequence:

But it is easier to use this Rule:
xn = n(n+1)/2

Example:
the 5th Triangular Number is x5 = 5(5+1)/2 = 15,
and the sixth is x6 = 6(6+1)/2 = 21

Square Numbers
1, 4, 9, 16, 25, 36, 49, 64, 81, ...
The next number is made by squaring where it is in the pattern.
Rule is xn = n2



Cube Numbers
1, 8, 27, 64, 125, 216, 343, 512, 729, ...
The next number is made by cubing where it is in the pattern.
Rule is xn = n3
Series
Patterns
Sequence
When you sum up just part of a sequence it is called a Partial Sum.
But when you sum up an infinite sequence it is called a "Series" (it sounds like another name for sequence, but it is actually a sum).
Example: Odd numbers
Sequence: {1,3,5,7,...}
Series: 1+3+5+7+...
Partial Sum of first 3 terms: 1+3+5
ARITHMETIC PROGRESSIONS (A.P.)
What is an Arithmetic Progression?
A sequence a1, a2, a3, a4,...,an,... is called an arithmetic progression, if there exists a constant number d such that
a2=a1+d
a3=a2+d
a4=a2+d
an=a(n-1)+d and so on...
The constant d is called the common difference of the A.P.
General Term of an A.P.
Let a be the first given term and d be the common difference of an A.P. Then, its nth term or general term is given by
an = a + (n-1)d
Sum of first n terms of an A.P.
Applications based questions on A.P.
For example,
1) 2, 5, 8, 11, 14, 17, 20, 23, 26, 29...
2)0, 10, 20, 30, 40, 50,60,70,80,90...
General term of an A.P. = First term + (term number-1) x (common difference)
Formulae

Sum of first n terms of an A.P. is
Sn = (n/2)[2a +(n -1)d]
If Sn Is sum of n terms of an A.P. whose first term is a and last term is l,
then Sn = (n/2)(a + l)
If common difference is d, number of terms n and the last term l,
then Sn = (n/2)[2l-(n -1)d]
Note:
Sometimes the question involves 3, 4 or 5 terms of an A.P.
If the sum of the numbers is given, then in an A.P.,
three numbers are taken as a -d, a, a +d
four numbers are taken as a -3d, a -d, a +d, a +3d.
five numbers are taken as a -2d, a -d, a, a +d, a +2d.
This considerably simplifies the calculations of a and d.
Example
Example

Sum the series 2 +4 +6 +... upto 40 terms.
Find the sum of first 19 terms of the A.P. whose nth term is 2n +1.
Find the sum of the series 1 +3 +5 +... +99.
Solution

We see that given series is an A.P. with first term a = 2,
common difference d = 2 and number of terms n = 40
Hence S40 = n[2a +(n -1)d]/2 = 40[2.2 +(40 -1)2]/2
= 20(4 +78) = 1640
Here first term a = T1 = 2n +1 = 2(1) +1 = 3,
and last term = T19 = 2(19) +1 = 39
S19 = n(a +l)/2 = 19(3 +39)/2 = 19.21 = 399
Given series is an A.P. with a = 1, l = 99, d = 2
To find the number of terms, we use l = a +(n -1)d
=> 99 = 1 +(n -1)2 => n -1 = 49 => n = 50
Sn = n(a +l)/2 = 50(1 +99)/2 = 50.50 = 2500

1) Sum of 0·7 +0·71 +0·72 +... upto 100 terms.
2)How many terms of the A.P. 17 +15 +13 +... must be taken so that sum is 72? Explain the double answer.

Solution:

1) We see that given series is an A.P. with first term
a = 0·7 and common difference d = 0·01
S100 = n[2a +(n -1)d]/2 = 100[2(0·7) +(100 -1)(0·01)]/2
= 50 (1· 4 +0·99) = 119·5

2) We are given a = 17, d = -2, Sn = 72 and we have to find n.
Using Sn = n[2a +(n -1)d]/2, we get
72 = n[2.17 +(n -1)(-2)]/2 = n(36 -2n)/2 = n(18 -n)
=> n² -18n +72 = 0 => (n -6)(n -12) = 0
=> n = 6, 12
Both values of n, being positive integers are valid. We get double answer because sum of 7th to 12th terms is zero, as some terms are positive and some are negative.
The interior angles of a polygon are in arithmetic progression. The smallest angle is 120° and the common difference is 5°. Find the number of sides in the polygon.
Solution

The sum of all exterior angles of a polygon = 360°. If the interior angles are in A.P., then exterior angles are also in A.P.; largest exterior angle is 180°-120° = 60° and common difference is -5°.
Sum = 360° = n [2 a +(n -1) d]/2 = [2. 60° +(n -1)(-5°)]/2
=> 720 = 120 n -5n² +5n => 5n² -125n +720 = 0
=> n² -25 n +144 = 0 => (n -9)(n -16) = 0
=> n = 9 or n = 16.
However if n = 16, then internal angles would vary from 120° to 120° +(16 -1)(5°) = 195°; so one of the internal angles will be 180° which is not possible in a polygon.
Hence the only correct solution is n = 9.
The sum of three numbers in A.P. is 18 and their product is 192. Find the numbers.
Solution

Let the three numbers in A.P. be a -d, a, a +d. By given conditions,
(a -d) +a +(a +d) = 18, (a -d)a(a +d) = 192
=> 3a = 18, a(a² -d²) = 192 => a = 6, 6(36 -d²) = 192
=> d² = 36 -192/6 = 36 -32 = 4 => d = ±2.
With a = 6, d = 2, we get three numbers as 4, 6, 8.
With a = 6, d = -2, we get three numbers as 8, 6, 4.
Hence the required numbers are 4, 6, 8.
If Sn denotes the sum of n terms of an ap whose common difference is d and the first term is a the find -
Sn - 2Sn-1 + Sn-2
Solution:
Given, a and d are the first term and common difference of the A.P.
Sum of n term of the A.P,

Patterns and designs with A.P.
Fibonacci's sequence in nature
1) You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total distance the object will fall in 6 seconds?

Ans)
2) The sum of the interior angles of a triangle is 180º, of a quadrilateral is 360º and of a pentagon is 540º. Assuming this pattern continues, find the sum of the interior angles of a dodecagon (12 sides).
Ans)
Q 3)
Ans)
Q)
Ans)
Geometric Progression (G.P.)
What Is Geometric Progression?
Sum of n terms of a G.P.
Application based questions
Examples:
Videos
Sequences & Series
A.P. & Sum of n terms in A.P.
G.P.
Full transcript