Method of differences

Calculate

• You may notice that a lot of terms cancel out:
  • is a negative term in the first bracket, and a positive term in the second bracket.
  • is a negative term in the second bracket, and a positive term in the third bracket.
  • This pattern continues for all the middle terms, the last being .
• This leaves us with just .
• Simplifying this gives us .
• Answer: .

This is called the ‘method of differences’!

Show that

• (write down the left-hand side)
• write each term with a common denominator
• (get a common denominator)
• (expand the numerator)
• (simplify)

Hence solve

• (use the result from above)
• 
  
  (write out the first and last few terms)
• All except the first and last terms cancel out, leaving us with
• (simplify)
• Answer: .

Given that , show that

• (write out the first few terms)
• (all middle terms cancel out)
• (expand the left-hand side)
• (rearrange)
• (expand the right-hand side)
• (simplify)
• (factorise out a 4n)
• (factorise the quadratic)
• (divide both sides by 24)
• (simplify)