1. Sigma notation The Greek letter Σ is used to denote summing If the terms of a sequence are then, for example, From the 1 st term To the 5 th If you are given the algebraic rule for u r, this could be evaluated: 25811 Eg evaluate

2. Gauss’s Genius 500 times A cruel schoolmaster once asked an errant pupil to add up the numbers from 1 to 500, thinking it would keep them the quiet for the rest of the day. What he couldn’t know was that the pupil was called Gauss, and would go on to become one of the greatest Mathematicians of all time. He produced an answer in a matter of minutes using a very clever method! This method can be applied to the sum of any arithmetic sequence

3. n times Consider the sum of the first n positive integers Eg find the value of 1 + 2 + 3 + … + 2000 This rule can be used to instantly work out the sum to any number!

4. Series summation formulae You have already seen that: It can also be shown that: Eg evaluate ‘Manually’Using formula It can also be shown that: Eg evaluate ‘Manually’Using formula

5. Adapting series summation formulae Eg Consider This is doubled In general, if a is a constant, Eg Write down an expression for

6. Adapting series summation formulae Eg Consider This is In general, Eg find an expression for

7. Eg Simplifying series summations Eg from previously Factorising can be extended to removing a fraction: Take the LCM of the denominators as the factor’s denominator Simplify the bracketed expression LCM of 6 and 2 is 6 Simplify Factorise again if possible

8. Evaluating series summations This can be found by doing Eg from previously Finding an algebraic expression for a summation enables you to evaluate it quickly Hence find Summations not starting from r = 1 can also be easily evaluated: Eg evaluate

9. WB29(a) Use the results for, and to prove that for all positive integers n. (b) Hence, or otherwise, find the value of

10. WB35(b) Using the formulae for and, show that (b) Hence evaluate

11. Adapting series summation formulae Eg from previously Hence find You can find an expression for the summation to any value by substituting it into the expression

12. WB36(b) Using the standard results for and, show that where a and b are integers to be found. (c) Hence show that

13. WB30(a) Use the results for and to show that for all positive integers n. (b) Hence show that Where a and b are integers to be found

14. Sigma notation and key series identities Key identitiesExample

15. Proving the rule for Σr 2