11.1 Sequences and Series | Arithmetic Progressions, Geometric Progressions, Further Sequences & Series

Arithmetic Sequences

What do I need to know about arithmetic progressions?

• In an arithmetic progression (also called arithmetic sequence), the difference between consecutive terms in the sequence is constant

• That constant difference is known as the common difference of the sequence
• You need to know the nth term formula for an arithmetic progression

• • a is the first term
  • d is the common difference

Exam Tip

• The formula is given in the formula booklet
• If you know two terms in an arithmetic sequence you can find a and d using simultaneous equations. 

Worked Example

Arithmetic Series

How do I find the sum of an arithmetic progression?

• The sum of the terms of an arithmetic progression is sometimes called an arithmetic series

• The following formulae will let you find the sum of the first n terms of an arithmetic progression:

   or   

• • a is the first term
  • d is the common difference
  • l is the last term

• You can use whichever formula is more convenient for a given question
• The a and the d in those formulae are exactly the same as the ones used with arithmetic progressions

 

How do I derive the formula for the sum of an arithmetic progression?

• Learn this proof of the sum of an arithmetic progression formula – you can be asked to give it on the exam:
  • Write the terms out once in order
  • Write the terms out again in reverse order
  • Add the two sums together
    • The terms will pair up to give the same sum 
    • There will be n of these terms
  • Divide by two as two of the sums have been added together

Exam Tip

The arithmetic series formulae are in the formulae booklet – you don't need to memorise them.

Worked Example

 

Geometric Sequences

What do I need to know about geometric progressions?

• In a geometric progression (also called geometric sequence) there is a common ratio between consecutive terms in the sequence

• You need to know the nth term formula for a geometric progression

• • a is the first term
  • r is the common ratio

Exam Tip

• The formula is given in the formula booklet
• If you know two terms in a geometric sequence you can find a and r using simultaneous equations. 

Worked Example

Geometric Series

How do I find the sum of a geometric progression?

• The sum of the terms of a geometric progression is sometimes called a geometric series

  

• The following formulae will let you find the sum of the first n terms of a geometric progression:

   or   

• • a is the first term
  • r is the common ratio

  

• The one on the left is more convenient if r < 1, the one on the right is more convenient if r > 1
• The a and the r in those formulae are exactly the same as the ones used with geometric progression

 

How do I prove the formula for the sum of a geometric progression?

• Learn this proof of the sum of a geometric progression formula – you can be asked to give it in the exam:
  • Write out the sum once
  • Write out the sum again but multiply each term by r
  • Subtract the second sum from the first
    • All the terms except the two should cancel out
  • Factorise and rearrange to make S the subject

 

What is the sum to infinity of a geometric series?

• If (and only if!) |r| < 1, then the sum of a geometric progression converges to a finite value given by the formula

  

• S∞ is known as the sum to infinity
• If |r| ≥ 1 the sum of a geometric progression is divergent and the sum to infinity does not exist

 

Worked Example

Language of Sequences & Series

What is a progression/sequence?

• A progression (or sequence) is an ordered set of numbers with a rule for finding all the numbers in the progression

  

• The numbers in a sequence are called terms
• The terms of a progression are often referred to by letters with a subscript

 

 

What is a series?

• You get a series by summing up the terms in a progression

 

• We use the notation Sn to refer to the sum of the first n terms in the progression

  ie.   Sn = u1 + u2 + u3 + … + un

Increasing, decreasing and periodic progressions

• A progression is increasing if un+1 > un for all positive integers n – ie if every term is greater than the term before it
• A progression is decreasing if un+1 < un for all positive integers n – ie if every term is less than the term before it

• A progression is periodic if the terms repeat in a cycle
• The order (or period) of a periodic progression is the number of terms in each repeating cycle

Exam Tip

Look out for progressions defined by trigonometric functions – this can be a way of 'hiding' a periodic function.

Worked Example

Sigma Notation

What is sigma notation?

• The symbol Σ is the capital Greek letter sigma – that's why it's called 'sigma notation'!
• 'Σ' stands for 'sum' – the expression to the right of the Σ tells you what is being summed, and the limits above and below tell you which terms you are summing

 

 

• Be careful – the limits don't have to start with 1!
  • For example:  or  

What do I need to be able to do with sigma notation?

• Sigma notation can be used to represent both arithmetic progressions and geometric progressions
  • Arithmetic will have the form 
  • Geometric will have the form 
  • Writing out the first few terms will help you

  

• To work out such a sum use the formulae for the sum of arithmetic and geometric progressions
• As long as the expressions being summed are the same you can add and subtract in sigma notation
  • For example:

 

Exam Tip

Be careful when more than one letter appears in a sigma notation question. 

Worked Example