Further Pure 1

1
Further Pure 1

• Summation of finite Series

2
Sigma notation

• In the last lesson we met the following rules.
• 1) 1 2 3 n (n/2)(n1)
• 2) 12 22 32 n2 (n/6)(n1)(2n1)
• 3) 13 23 33 n3 (n2/4) (n1)2
• We can write long summations like the ones above
  using sigma notation.

3
Sigma notation

• The r acts as a counter starting at 1 (or
  whatever is stated under the sigma sign) and
  running till you get to n (on top of the sigma
  sign).
• Each r value generates a term and then you simply
  add up all the terms.
• The terms in the example above come from
• r 1 211 3
• r 2 221 5
• r 3 231 7
• r 4 241 9
• The 4 on top of the sigma sign tells us to stop
  when r 4.

4
Questions

• Here are some questions for you to try and find
  the values of.

5
Sigma notation

• We can now remember the identities that we met
  last lesson and have mentioned already adding the
  sigma notation.

6
Using Nth terms

• Use the nth term to find the following summation.
• The summation only works if you sum from 1 to n.
• How would you calculate the next example.
• Here the sum goes from r 4, to r 8.
• This means you do not want the terms for r 1, 2
  3.
• So the answer will be the sum to 8 minus the sum
  to 3.

7
Rules of summing series

• Here are 2 rules that you need to be familiar
  with.
• There is a numerical example followed by a
  general rule
• k and a represent random constants.

8
Example

• These results can be used to find the sum to n of
  lots of different series.

• First break the summation up.
• Next use the general formula.
• Here (n/4)(n1) is a factor
• Next just multiply out and collect up like terms.
• Finally the expression will factorise.

9
Question

• Try this question

10
Question

• Here the sum starts at r 6.
• This is not as complicated as it may seem.
• All you need to do is take of the first 5 terms.
• So the sum from 6 to 30 is the sum to 30 minus
  the sum to 5.

11
Questions

• Here are some questions for you to find the nth
  terms of.
• The solutions are on the next two slides

12
Solutions
13
Solutions
14
Summation of a finite Series

• When Carl Friedrich Gauss was a boy in elementary
  school his teacher asked his class to add up the
  first 100 numbers.
• S100 1 2 3 100
• Gauss had a flash of mathematical genius and
  realised that the sum had 50 pairs of 101
• Therefore S100 50 101
• 5 050
• From this we can come up with the formula for the
  sum of the first n numbers.
• Sn (n/2)(n1)
• We have met this result a few times already.

15
Method of differences

• We can prove the same result using a different
  method.
• The method of differences.

16
Example 1

• Use the method of differences to find the sum to
  30 of the following example.
• Solution to part ii is on the next 2 slides.
• You covered adding fractions in C2 and should be
  able to get the answer.

17
Example 1

• We can use the identity to
  re-arrange the question.
• Now write the summation out long hand. Starting
  with r 1.
• Then r 2,3 etc.
• Write out the last 2 or 3 terms.
• Having written out the full summation you can
  spot that parts of the sum cancel.
• The bits that are left do not cancel and we can
  sort out the sum.

18
Example 2

• In this next example we will find the sum to n.

• Solution to part ii is on the next 2 slides.
• You covered adding fractions in C2 and should be
  able to get the answer.

19
Example 2

• We can use the identity to
  re-arrange the question.
• Now write the summation out long hand. Starting
  with r 1.
• Then r 2,3 etc.
• Write out the last 3 terms.
• Having written out the full summation you can
  spot that parts of the sum cancel.
• The bits that are left do not cancel and we can
  sort out the algebra.