International Baccalaureate Mathematics

Syllabus Content

Sum of infinite convergent geometric sequences.

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Geometric Sequences An exercise on geometric sequences including finding the nth term and the sum of any number of terms.

Here is an exam-style questions on this statement:

"In a geometric series the common ratio is $r$ and sum to $n$ terms is $S_n$." ... more

Here is an Advanced Starter on this statement:

Grandmother
How far would grandma have travelled after a suitably large number of days given her walking regime? more

Click on a topic below for suggested lesson Starters, resources and activities from Transum.

Sequences A pattern of numbers following a rule is called a sequence. There are many different types of sequence and this topic introduces pupils to some of them. The most basic sequences of numbers is formed by adding a constant to a term to get the next term of the sequence. This rule can be expressed as a linear equation and the terms of the sequence when plotted as a series of coordinates forms a straight line. More complex sequences are investigated where the rule is not a linear function. Other well-known sequences includes the Fibonacci sequence where the rule for obtaining the next term depends on the previous two terms. Sequences can be derived from shapes and patterns. A growing patterns of squares or triangles formed from toothpicks is often used to show linear sequences in a very practical way. Diagrams representing sequences provides interesting display material for the classroom. Typically pupils are challenged to find the next term of a given sequence but a deeper understanding is needed to find intermediate terms, 100th term or the nth term of a sequence.

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Official Guidance, clarification and syllabus links:

Use of $|r|<1$ and modulus notation.

Link to: geometric sequences and series (SL1.3).

Formula Booklet:

The sum of an infinite geometric sequence

$ S_\infty = \dfrac{u_1}{1-r}, \quad |r| \lt 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. If the common ratio is greater than -1 and less than 1, the sequence is said to be convergent. The sum of an infinite convergent geometric sequence can be found using a specific formula. This sum is finite and is often referred to as the limit of the sequence as the number of terms approaches infinity.

The formula to find the sum $ S $ of an infinite convergent geometric sequence with a first term $ a $ and a common ratio $ r $ (where $ |r| < 1 $) is given by:

$$ S = \frac{a}{1 - r} $$

For example, consider an infinite geometric sequence with a first term $ a = 3 $ and a common ratio $ r = 0.5 $.
Using the formula, the sum of this sequence is:

$$ S = \frac{3}{1 - 0.5} = \frac{3}{0.5} = 6 $$

Thus, the sum of the infinite convergent geometric sequence with terms starting at 3 and halving each time is 6.

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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International Baccalaureate Mathematics

Number and Algebra

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