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?

• • 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:

• • 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
• 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

• • This is?in the formula book, you do not need to remember it