Chapter 4 Mathematical Induction Used to verify a property of a sequence 2 4 6 8 for i 1 ai 2i infinite sequence with infinite distinct values for i 1 bi 1 i infinite sequence with finite distinct values for 1 i 6 ci i 5 finite sequence with finite distinct values Finding the Explicit Formula Figure the formula of this sequence 1 1 1 1 1 4 9 16 25 Different sequences with same initial values k 0 ak 2 k 1 bk k 1 3 k 2 1 Summation Product Notation Sum of Items Specified 6 k 1 2 k 21 2 2 2 3 2 4 2 5 2 6 Product of Items Specified 5 2 k 2 1 2 2 2 3 2 4 2 5 k 1 Variable ending point n as the index of the final term n k 0 k 1 n k for n 2 for n 3 2 Nesting of Sum Product Notation Variations same or different J n j j 1i 1 J Y ij 2 nj Yij 2 j 1 i 1 J nj Yij 2 j 1i 1 Telescoping Series n k k 1 k 1 k 1 k 2 n i i 1 i 1 3 Properties Merging and Splitting n ak k m n k m ak n bk k m n k m n ak bk k m bk n ak k m n ak bk k m n k m i ak k m ak i k m n ak k i 1 ak n ak k i 1 Distribution c n ak k m n c a k k m Factorial n n n 1 n 2 2 1 Definition 0 1 n n n 1 4
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