\( \DeclareMathOperator{cosec}{cosec} \)

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International Baccalaureate Mathematics

Number and Algebra

Syllabus Content

Geometric sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences

Use of sigma notation for the sums of geometric sequences.


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Official Guidance, clarification and syllabus links:

Spreadsheets, GDCs and graphing software may be used to generate and display sequences in several ways.

If technology is used in examinations, students will be expected to identify the first term and the ratio.

Link to: models/functions in topic 2 and regression in topic 4.

Examples include the spread of disease, salary increase and decrease and population growth.

Formula Booklet:

The nth term of a geometric sequence

\( u_n=u_1r^{n-1}\)

The sum of n terms of a finite geometric sequence

\( S_n = \dfrac{u_1(r^n-1)}{r-1} = \dfrac{u_1(1-r^n)}{1-r}. \quad r \neq 1\)

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted by \( r \).

The \( n^{th} \) term of a geometric sequence can be calculated using the formula:

$$ u_n = u_1 \cdot r^{(n - 1)} \\ \text{where } u_n \text{ is the } n^{th} \text{ term, } u_1 \text{ is the first term, and } r \text{ is the common ratio.} $$

The sum of the first \( n \) terms (\( S_n \)) of a geometric sequence can be found using the formula:

$$ S_n = \frac{u_1(1 - r^n)}{1 - r} \text{, for } r \neq 1 \\ \text{where } S_n \text{ is the sum of the first } n \text{ terms.} $$

Sigma notation (\( \Sigma \)) is also used to represent the sum of geometric sequences. For a geometric sequence, it can be represented as:

$$ S_n = \sum_{k=0}^{n-1} u_1 \cdot r^k $$

Example 1: Consider a geometric sequence where the first term \( u_1 = 2 \) and the common ratio \( r = 3 \). Let's find the \( 5^{th} \) term and the sum of the first 5 terms.

Using the formula for the \( n^{th} \) term:

$$ u_5 = 2 \cdot 3^{(5 - 1)} = 2 \cdot 81 = 162 $$

So, the \( 5^{th} \) term is 162.

Now, using the formula for the sum of the first \( n \) terms:

$$ S_5 = \frac{2(1 - 3^5)}{1 - 3} = \frac{2(1 - 243)}{-2} = \frac{484}{2} = 242 $$

Thus, the sum of the first 5 terms of the given geometric sequence is 242.

Example 2: For a geometric sequence with the first term \( u_1 = 4 \) and the common ratio \( r = 0.5 \), we can represent the sum of the first 4 terms using sigma notation as:

$$ S_4 = \sum_{k=0}^{3} 4 \cdot (0.5)^k $$

This notation concisely represents the sum of the sequence: 4, 2, 1, and 0.5, which equals 7.5.

If you use a TI-Nspire GDC there are instructions here for generating a sequence on the calculator.

This video on Geometric Sequences is from Revision Village and is aimed at students taking the IB Maths Standard level course.

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