FP1 MEI Induction and Series
For standard results I have left gaps before the equals sign as greek letters cannot be typed. From top to bottom they should be the sum of: 1, r, r^2, r^3 between r=1 and n. Where a number is below an expression such as the standard results, the expression is divided by this number. Finally, for the last standard result, the n is n^2 and the (n+1) is (n+1)^2.
I have left these blank as I prefer adding what cannot be typed in pen, after printing. Sorry if you are not printing.
- Created by: Chloe
- Created on: 21-04-14 13:21
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- Induction and Series
- Proof by Induction
- Step 2: Prove that if the result is true for n=k then it is true for n=k+1
- Step 3: conclude the result is true for n>=1
- Step 1: Prove the result is true for n=1
- Step 2: Prove that if the result is true for n=k then it is true for n=k+1
- Step 3: conclude the result is true for n>=1
- Step 2: Prove that if the result is true for n=k then it is true for n=k+1
- Proves a general expression for the sum of a series
- I.e. given a sum 1+2+...+n and an expression = n(n+1) 2
- Given an iterative formula and a term i.e. a=2
- Prove true for 1st term a. Prove true for (k+1)th term when true for the kth term
- Step 2: Prove that if the result is true for n=k then it is true for n=k+1
- Summation of series
- Arithmetic
- ak=a+(k-1)d where a is 1st term and d is the common difference
- Sum of 1st n terms: n(2a+(n-1)d) 2
- Geometric
- ak=ar^(k-1) where a is 1st term and r is the common ratio
- Sum of 1st n terms: a(1-r^n) (1-r)
- Method of differences
- Aim to split the given expression so that terms cancel
- Questions often split apart into parts i) show that.. ii) hence find...
- standard results
- =n(n+1)(2n+1) 6
- =n(n+1) 2
- =n (n+1) 4
- =n
- Arithmetic
- Proof by Induction
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