Geometric Sequences
What is a geometric sequence?
- In a geometric sequence, there is a common ratio, r, between consecutive terms in the sequence
- For example, 2, 6, 18, 54, 162, … is a sequence with the rule ‘start at two and multiply each number by three’
- The first term, u1, is 2
- The common ratio, r, is 3
- For example, 2, 6, 18, 54, 162, … is a sequence with the rule ‘start at two and multiply each number by three’
- A geometric sequence can be increasing (r > 1) or decreasing (0 < r < 1)
- If the common ratio is a negative number the terms will alternate between positive and negative values
- For example, 1, -4, 16, -64, 256, … is a sequence with the rule ‘start at one and multiply each number by negative four’
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- The first term, u1, is 1
- The common ratio, r, is -4
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- For example, 1, -4, 16, -64, 256, … is a sequence with the rule ‘start at one and multiply each number by negative four’
- Each term of a geometric sequence is referred to by the letter u with a subscript determining its place in the sequence
How do I find a term in a geometric sequence?
- The term formula for a geometric sequence is given as
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- Where is the first term, and is the common ratio
- This formula allows you to find any term in the geometric sequence
- It is given in the formula booklet, you do not need to know how to derive it
- Enter the information you have into the formula and use your GDC to find the value of the term
- Sometimes you will be given a term and asked to find the first term or the common ratio
- Substitute the information into the formula and solve the equation
- You could use your GDC for this
- Substitute the information into the formula and solve the equation
- Sometimes you will be given two or more consecutive terms and asked to find both the first term and the common ratio
- Find the common ratio by dividing a term by the one before it
- Substitute this and one of the terms into the formula to find the first term
- Sometimes you may be given a term and the formula for the nth term and asked to find the value of n
- You can solve these using logarithms on your GDC
Exam Tip
- You will sometimes need to use logarithms to answer geometric sequences questions
- Make sure you are confident doing this
- Practice using your GDC for different types of questions
Worked Example
The sixth term, , of a geometric sequence is 486 and the seventh term, , is 1458.
Find,
i)
the common ratio, , of the sequence,
ii)
the first term of the sequence, .
Geometric Series
How do I find the sum of a geometric series?
- A geometric series is the sum of a certain number of terms in a geometric sequence
- For the geometric sequence 2, 6, 18, 54, … the geometric series is 2 + 6 + 18 + 54 + …
- The following formulae will let you find the sum of the first n terms of a geometric series:
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- is the first term
- is the common ratio
- Both formulae are given in the formula booklet, you do not need to know how to derive them
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- You can use whichever formula is more convenient for a given question
- The first version of the formula is more convenient if and the second is more convenient if
- A question will often give you the sum of a certain number of terms and ask you to find the value of the first term, the common ratio, or the number of terms within the sequence
- Substitute the information into the formula and solve the equation
- You could use your GDC for this
- Substitute the information into the formula and solve the equation
Exam Tip
- The geometric series formulae are in the formula booklet, you don't need to memorise them
- Make sure you can locate them quickly in the formula booklet
Worked Example
A geometric sequence has and . Find the value of and .