Sequences and Series

Sequences and Series

Sequences and series play essential roles in mathematics and also in our everyday experiences. A sequence, often referred to as a progression, serves as the foundation for constructing a series. Sequences and series is the one of the basic concepts in IB Math SL AA program. Sequences and series have specific rules. These rules provide to arrange formulas for sequence and series. These formulae are given in IB Math SL DataBooklet. (data booklet a link verebiliriz)

The sequence is the group of numbers arranged in a specific order or rules. Series is formed by adding the terms in a sequence.

In IB Math AA syllables sequence and series on seen at

IB Math SL AA Topic 1 Arithmetic Sequences and Series

IB Math SL AA Topic 1 Geometric Sequence and Series

IB Math HL AA Topic 1 Arithmetic Sequence and Series

IB Math HL AA Topic 1 Geometric Sequence and Series

IB Math HL AA Topic 5 Harmonic Sequence and Series

IB Math HL AA Topic 5 Taylor and Maclaurin Series

Also Fibonacci series is given for daily life experiences.

Arithmetic Sequences

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers in which the difference between successive terms remains constant.

An arithmetic sequence can be represented as:

u₁, u₁ + d, u₁ + 2d, u₁ + 3d, ...

Where

u₁ is first term

d is common difference

For example, the sequence

3, 7, 11, 15, 19 …..

has a common difference 4, means each term increase by 4. Any term can easily find by formula u_n = u₁ + (n-1)d.

Consider investigating eduib key concepts. (link verebiliriz)

If sequence is finite, sum of the terms can be found by Gauss Sum.

Sum of first term and last term will be multiplied by number of terms and then divided by 2.

Sum of sequence is given by S_n=n/2(u₁+u_n)

These formulas are written in IB Math AA SL formula data booklet. (data booklet yükleyebiliriz)

EduIB Questionbank gives you opportunity to solve real past paper like questions about arithmetic sequence. (sorulara link verebiliriz)

For example one of arithmetic sequence question in EduIB is shown below. It is very similar to IB Math SL Exam in November 2014.

Geometric Sequence

A geometric sequence, also known as a geometric progression, is a sequence of numbers in which the ratio between successive terms remains constant.

A geometric sequence can be represented as:

u₁ ,  u₁.r ,  u_1.r², u_1.r³ , …

where

u₁ is first term

r is common ratio

For example, the sequence

3, 12, 48, 192, ….

has a common ratio 4, means each terms found by multiplying the preceding term by 4. Any term can easily find by formula u_n = u₁.r^(n-1).

Consider investigating eduib key concepts. (link verebiliriz)

If sequence is finite sum can be found by formula S_n= u_1.(rⁿ-1)/(r-1).

If sequence is infinite and absolute value of r is less than one then sum is equal u₁/(r-1).

If absolute value of ratio is greater than 1 than sum of infinite sequence may be positive infinity or negative infinity.

These formulas are written in IB Math AA SL formula data booklet. (data booklet yükleyebiliriz)

EduIB Questionbank gives you opportunity to solve real past paper like questions about geometric sequence. (sorulara link verebiliriz)

For example one of arithmetic sequence question in EduIB is shown below. It is very similar to IB Math SL Exam in May 2016.

Tips for Understanding Sequences and Series:

1) Arithmetic Sequence and Series Notation: In an arithmetic sequence and series, denote the first term as "a", the common difference as "d", the nth term as "an", and the number of terms as "n".

2) Representation of Arithmetic Sequence:  An arithmetic sequence can generally be represented as: a, a+d, a+2d, a+3d, and so on, where each successive term differs by the common difference "d"

3) Successive Terms in Geometric Sequence and Series: Each subsequent term in a sequence forms a geometric progression, where it is obtained by multiplying the common ratio to its preceding term.

4) Formula for nth term in Geometric Sequence: The formula for finding the nth term (a_n) in a geometric progression with the first term "a" and

common ratio "r" is: a_n = ar^(n-1).

5) Sum of Infinite Geometric Series: The sum of an infinite geometric progression can be calculated using the formula: S_n = a₁ / (1 - r), where the absolute value of the common ratio "r" is less than 1 (|r| < 1).