Summation

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Summation

• Using a method of differences to sum a series

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Reminder from FP1
Find the nth term of the following summations.
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Solutions
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Solutions
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Method of differences

• This method is used when we can transform a
  summation into two parts which cancel as the
  terms are added.
• This will be explained further in the following
  slide.
• The examples which follow it will make things a
  little clearer.

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Method of differences

• If I wanted to sum the following

• It may be possible to express it as..

• So when I sum the terms of ur, I get..

As many parts of this will cancel, Im just left
with a simple expression for the sum of the
series.
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Example 1
Use the method of differences to find the sum to
30 of the following example.
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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.

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Example 2
In this next example we will find the sum to n.
i) Show that
ii) Hence find
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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.