Geometric Sequences
What is a geometric progression?
- In a geometric progression (also called geometric sequence) there is a common ratio, r, between consecutive terms in the sequence
- For example, 2, 6, 18, 54, 162, … is a progression with the rule ‘start at two and multiply each number by three’
- The first term, a, is 2
- The common ratio, r, is 3
- For example, 2, 6, 18, 54, 162, … is a progression with the rule ‘start at two and multiply each number by three’
- A geometric progression 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, a, 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 progression 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 progression?
- The term formula for a geometric progression 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 progression
- It is given in the list of formulas, you do not need to know how to derive it
- Enter the information you have into the formula and 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
- 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
Examiner Tip
- The formula is given in the list of formulas
- If you know two terms in a geometric progression you can find a and r using simultaneous equations