Edexcel A Level Further Maths: Core Pure:复习笔记3.2.1 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
  • 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 ( ) 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  • This is important – if the top and bottom limits don’t all match then the relation is no longer valid!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 as 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,