Sums of Integers, Squares & Cubes (Edexcel A Level Further Maths): Revision Note

Sums of Integers, Squares & Cubes

How can we use sigma notation?

• When writing the sum of a series you can use sigma notation

  • The series  can instead be written as  or 

  • This means “the sum of all the terms from to for the sequence described by "

  • would mean 

• Using the following relations, summations can be grouped together (or ungrouped) to make some calculations easier:

• a and b are constants, and and  are a functions of r

• Note that the top and bottom summation limits ( and ) are the same for all the sums

  • This is important – if the top and bottom limits don’t all match then the relation is no longer valid!

• This can be very useful when or or , as and  (or or ) are straightforward to find using formulae

• A useful result to remember is that 

What are the formulae for finding sums of integers, squares, and cubes?

• There are several useful formulae for summing integers, square numbers, and cube numbers

• The sum of the first natural numbers is given by 

  • 

    • This formula is not given in the formula book

• The sum of the first square numbers is given by

  •  

    • This formula is given in the formula book

• The sum of the first cube numbers is given by

  •  

    • This formula is given in the formula book

    • Notice that this is equal to the formula for the sum of the first natural numbers, squared

• Using the relations given above, a more complicated summation can often be broken down into sums of constants, natural numbers, squares, and cubes

  • For example,