# 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.

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• Created by: Chloe
• Created on: 21-04-14 13:21
• 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
• 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
• 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