Section 2

Eng/Ahmed Rashed
Eng/Nagwa Mohamed
 Revision on Sequences
Q1) Find a rule for the following sequence:
{17, 21, 25, 29, 33, …}
 Revision on Sequences
Q1) Find a rule for the following sequence:
{17, 21, 25, 29, 33, …}

• Answer:
The sequence is an arithmetic sequence since
𝒂𝟐 − 𝒂𝟏 = 𝟐𝟏 − 𝟏𝟕 = 𝟒
𝒂𝟑 − 𝒂𝟐 = 𝟐𝟓 − 𝟐𝟏 = 𝟒
𝒂𝟒 − 𝒂𝟑 = 𝟐𝟗 − 𝟐𝟓 = 𝟒
So, we can use the rule for the arithmetic sequence:
𝒙𝒏 = 𝒂 + 𝒅(𝒏 − 𝟏)
Where a is the first term and d is the common difference.
 Revision on Sequences
Q1) Find a rule for the following sequence:
{17, 21, 25, 29, 33, …}

• Answer:
a = 17
d=4
So, the rule is: 𝒙𝒏 = 𝟏𝟕 + 𝟒(𝒏 − 𝟏)
 Revision on Sequences
Q2) Find a rule for the following sequence:
{310, 2170, 15190, 106330, 744310, …}
 Revision on Sequences
Q2) Find a rule for the following sequence:
{310, 2170, 15190, 106330, 744310, …}

• Answer:
The sequence is a geometric sequence since
𝒂𝟏 /𝒂𝟐 = 𝟕
𝒂𝟑 /𝒂𝟐 = 𝟕
𝒂𝟒 /𝒂𝟑 = 𝟕
So, we can use the rule for the geometric sequence:
𝒙𝒏 = 𝒂𝒓𝒏−𝟏
Where a is the first term and r is the common ratio.
 Revision on Sequences
Q2) Find a rule for the following sequence:
{310, 2170, 15190, 106330, 744310, …}

• Answer:
a = 310
r=7
So, the rule is: 𝒙𝒏 = 𝟑𝟏𝟎 ∗ 𝟕𝒏−𝟏
Summation (Sigma)
 Summation (Sigma)
• Evaluate the following summation:
 Summation (Sigma)
• Evaluate the following summation:

= 𝟏𝟐 + 𝟐𝟐 + 𝟑𝟐 + 𝟒𝟐
= 𝟏 + 𝟒 + 𝟗 + 𝟏𝟔
= 𝟑𝟎
 Summation (Sigma)
• Evaluate the following summation:
 Summation (Sigma)
• Evaluate the following summation:

𝟑 𝟒 𝟓
= + +
𝟑+𝟏 𝟒+𝟏 𝟓+𝟏

𝟑 𝟒 𝟓
= + +
𝟒 𝟓 𝟔
𝟏𝟒𝟑
=
𝟔𝟎
 Summation Properties
• Multiplying by a constant:
• We can pull the constant outside the sigma.
 Summation Properties
• Adding or Subtracting:
• We can split the summation into two summations.
 Summation Shortcuts
1.

2.

3.
 Summation
• Evaluate the following summation:
 Summation
• Evaluate the following summation:

• Answer:
 Summation
• Evaluate the following summation:
 Summation
• Evaluate the following summation:
• Answer:
 Series
• Series: the sum of elements of a sequence.
• For example, if {5, 10, 15, 20, 25, …} is a sequence
• Then 5 + 10 + 15 + 20 + 25 + … is a series.

• Partial sum: the sum of part of the sequence.

• For example, if {5, 10, 15, 20, 25, …} is a sequence
• Then the sum of the first three elements (5 + 10 + 15 = 30) is a partial
sum.
 Partial Sum: Arithmetic Sequence
• For any arithmetic sequence, the sum of the first n terms can be
calculated as follows:
𝒏
𝒔𝒏 = (𝒂𝟏 + 𝒂𝒏 )
𝟐
• Where n is the number of terms, 𝒂𝟏 is the first term, 𝒂𝒏 is the n-th term.
 Partial Sum: Arithmetic Sequence
• Example: Find the 50𝑡ℎ partial sum of the arithmetic sequence
{-6, -2, 2, 6, …}
 Partial Sum: Arithmetic Sequence
• Example: Find the 50𝑡ℎ partial sum of the arithmetic sequence
{-6, -2, 2, 6, …}
• Answer: We need to find 𝒔𝟓𝟎 .
We have 𝒏 = 𝟓𝟎, 𝒂𝟏 = −𝟔, we need to find 𝒂𝟓𝟎 . We can find it using
the arithmetic sequence rule:
𝒙𝒏 = 𝒂 + 𝒅(𝒏 − 𝟏)
 Partial Sum: Arithmetic Sequence
• Example: Find the 50𝑡ℎ partial sum of the arithmetic sequence
{-6, -2, 2, 6, …}
• Answer: We need to find 𝒔𝟓𝟎 .
We have 𝒏 = 𝟓𝟎, 𝒂𝟏 = −𝟔, we need to find 𝒂𝟓𝟎 . We can find it using
the arithmetic sequence rule:
𝒙𝒏 = 𝒂 + 𝒅(𝒏 − 𝟏)
• a = -6, d = 4, so:
𝒙𝒏 = −𝟔 + 𝟒 𝒏 − 𝟏
𝒙𝟓𝟎 = −𝟔 + 𝟒 𝟓𝟎 − 𝟏 = 𝟏𝟗𝟎
𝟓𝟎
• So, 𝒔𝟓𝟎 = −𝟔 + 𝟏𝟗𝟎 = 𝟒𝟔𝟎𝟎.
𝟐
 Partial Sum: Arithmetic Sequence
• Example: find the 25𝑡ℎ partial sum of the arithmetic sequence
{17, 21, 25, 29, 33, …}
 Partial Sum: Arithmetic Sequence
• Example: find the 25𝑡ℎ partial sum of the arithmetic sequence
{17, 21, 25, 29, 33, …}
• Answer: We need to find 𝒔𝟐𝟓 .
We have 𝒏 = 𝟐𝟓, 𝒂𝟏 = 𝟏𝟕, we need to find 𝒂𝟐𝟓 . We can find it using
the arithmetic sequence rule:
𝒙𝒏 = 𝒂 + 𝒅(𝒏 − 𝟏)
 Partial Sum: Arithmetic Sequence
• Example: find the 25𝑡ℎ partial sum of the arithmetic sequence
{17, 21, 25, 29, 33, …}
• Answer: We need to find 𝒔𝟐𝟓 .
We have 𝒏 = 𝟐𝟓, 𝒂𝟏 = 𝟏𝟕, we need to find 𝒂𝟐𝟓 . We can find it using
the arithmetic sequence rule:
𝒙𝒏 = 𝒂 + 𝒅(𝒏 − 𝟏)
• a = 17, d = 4, so:
𝒙𝒏 = 𝟏𝟕 + 𝟒 𝒏 − 𝟏
𝒙𝟐𝟓 = 𝟏𝟕 + 𝟒 𝟐𝟓 − 𝟏 = 𝟏𝟏𝟑
𝟐𝟓
• So, 𝒔𝟐𝟓 = 𝟏𝟕 + 𝟏𝟏𝟑 = 𝟏𝟔𝟐𝟓.
𝟐
 Partial Sum: Geometric Sequence
• For any geometric sequence, the sum of the first n terms can be
calculated as follows:
𝟏 − 𝒓𝒏
𝒔𝒏 = 𝒂 𝟏 ( )
𝟏−𝒓
• Where n is the number of terms and 𝒂𝟏 is the first term.
 Partial Sum: Geometric Sequence
• Example: find the 7𝑡ℎ partial sum of the geometric sequence
{1, 2, 4, 8, 16, …}
 Partial Sum: Geometric Sequence
• Example: find the 7𝑡ℎ partial sum of the geometric sequence
{1, 2, 4, 8, 16, …}
• Answer: We need to find 𝒔𝟕 .
We have n = 7, 𝒂𝟏 = 𝟏, r = 2
𝟏−𝟐𝟕
So, 𝒔𝟕 = 𝟏 = 𝟏𝟐𝟕
𝟏−𝟐
 Partial Sum: Geometric Sequence
• Example: find the 15𝑡ℎ partial sum of the geometric sequence
{310, 2170, 15190, 106330, 744310, …}
 Partial Sum: Geometric Sequence
• Example: find the 15𝑡ℎ partial sum of the geometric sequence
{310, 2170, 15190, 106330, 744310, …}
• Answer: We need to find 𝒔𝟏𝟓 .
We have n = 15, 𝒂𝟏 = 𝟑𝟏𝟎, r = 7
𝟏−𝟕𝟏𝟓
So, 𝒔𝟏𝟓 = 𝟑𝟏𝟎 = 𝟐𝟒𝟓𝟐𝟗𝟎𝟔𝟕𝟖𝟎𝟏𝟑𝟔𝟕𝟎
𝟏−𝟕
 Matrix
Matrix :
A rectangular arrangement of 𝒎 × 𝒏 numbers, in m rows and n columns and enclosed within a bracket.

• The numbers are called the elements or the entries of the matrix.
• The horizontal lines of elements are said to constitute rows of the matrix
• the vertical lines of elements are said to constitute columns of the matrix.
• A matrix is denoted by a bold capital letter and the elements within the matrix are denoted by lower case
letters e.g. matrix [A] with elements aij
 Order of a Matrix
A matrix having m rows and n columns is called a matrix of order 𝒎 × 𝒏 or simply 𝒎 × 𝒏 matrix (read
as an m by n matrix).
 Types of Matrices
Column matrix or vector:
The number of rows may be any integer but the number of columns is always 1

Example:-
 Types of Matrices
Row matrix or vector
Any number of columns but only one row

Example:-
 Types of Matrices
Square matrix
The number of rows is equal to the number of columns
(a square matrix A has an order of m)

Example:-
 Types of Matrices
Rectangular matrix
has an unequal number of rows and columns and hence the order of a rectangular matrix
is of the form 𝒎 × 𝒏

Example:-
 Types of Matrices
Unit or Identity Matrix
An Identity Matrix has 1s on the main diagonal and 0s everywhere else.

Example:- A 3×3 Identity Matrix

• It is square (same number of rows as columns)

• Its symbol is the capital letter I
• It is the matrix equivalent of the number "1", when we multiply with it the original is unchanged:

A×I=A

I×A=A
 Types of Matrices
Diagonal Matrix
A diagonal matrix has zero anywhere not on the main diagonal

Example:-
 Types of Matrices
Scalar Matrix
A scalar matrix has all main diagonal entries the same, with zero everywhere else.

Example:-
 Types of Matrices
Triangular Matrix
Lower triangular is when all entries above the main diagonal are zero:

A lower triangular matrix

Upper triangular is when all entries below the main diagonal are zero:

An upper triangular matrix

 Types of Matrices
Null (zero) matrix - 0
All elements in the matrix are zero

Example:-
 Matrix Operations
Adding
To add two matrices: add the numbers in the matching positions.

Example:-

These are the calculations:

3+4=7 8+0=8
4+1=5 6−9=−3

• The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size.
 Matrix Operations

Subtracting
To subtract two matrices: subtract the numbers in the matching positions:
Example:-

These are the calculations:

3−4=−1 8−0=8
4−1=3 6−(−9)=15
 Matrix Operations

SCALAR MULTIPLICATION OF MATRICES

Matrices can be multiplied by a scalar (constant or single element)

Example:-

These are the calculations:

2×4=8 2×0=0
2×1=2 2×−9=−18
 Matrix Operations

Multiplying by Another Matrix

• The product of two matrices is another matrix

• Two matrices A and B must be conformable for multiplication to be possible
• i.e. the number of columns of A must equal the number of rows of B
• To multiply an m×n matrix by an n×p matrix, the ns must be the same, and the result is
an m×p matrix.

Example:-
A x B = C
(1x3) (3x1) (1x1)
 Matrix Operations

Multiplying by Another Matrix

to multiply a matrix by another matrix we need to do the "dot product" of rows and columns
 Matrix Operations

Multiplying by Another Matrix

Assuming that matrices A, B and C are conformable for the operations indicated, the
following are true:
1. AI = IA = A
2. A(BC) = (AB)C = ABC (associative law)
3. A(B+C) = AB + AC (first distributive law)
4. (A+B)C = AC + BC (second distributive law)
5. AB not generally equal to BA, BA may not be conformable
6. If AB = 0, neither A nor B necessarily = 0
7. If AB = AC, B not necessarily = C
 Matrix Operations

Transpose Matrix
The Transpose Matrix is obtained by changing its rows into columns and its columns
into rows

Example:-
The main diagonal stays the same.
 ‫!‪Thank you‬‬

‫ﷺصلّوا علي النبي‬