1.3.3 Geometric Sequences & Series

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
• 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’
    • • The first term, u₁, is 1
      • The common ratio, r, is -4
• 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

• • 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
• 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 n^th term and asked to find the value of n
  • You can solve these using logarithms on your GDC

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:

• • • 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
• 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

What is the sum to infinity of a geometric series?

• A geometric sequence will either increase or decrease away from zero or the terms will get progressively closer to zero
  • Terms will get closer to zero if the common ratio, r, is between 1 and -1
• If the terms are getting closer to zero then the series is said to converge

• • This means that the sum of the series will approach a limiting value
  • As the number of terms increase, the sum of the terms will get closer to the limiting value

How do we calculate the sum to infinity?

• If asked to find out if a geometric sequence converges find the value of r
  • If   then the sequence converges
  • If then the sequence does not converge and the sum to infinity cannot be calculated
  • means 
• If , then the geometric series converges to a finite value given by the formula

• • is the first term
  • is the common ratio
  • This is in the formula book, you do not need to remember it