am_lect_17 - Example Claim: n , n(n + 1) i = 2 i =1 n 0...

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Example Claim : Proof by induction on n : Base case ( n = 0): We need to show that P (0) is true i.e., Now LHS = 0 (empty sum) and RHS = 0. Hence P (0) is true Inductive step: Let k N such that P ( k ) is true i.e., We need to show that Now LHS = = RHS Hence P ( k +1) is true. The proof is now complete by induction. 2 ) 1 ( , 1 + = Ν = n n i n n i 2 ) 1 0 ( 0 0 1 + = = i i 2 ) 1 ( 1 + = = k k i k i 2 ) 1 ) 1 )(( 1 ( 1 1 + + + = + = k k i k i ) 1 ( 1 1 1 + + = = + = k i i k i k i 2 ) 1 ) 1 )(( 1 ( ) 1 ( 2 ) 1 ( + + + = + + + = k k k k k
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Another Example Claim : n N , 3 ( n 3 n ) Proof by induction on n : Base case ( n = 0): We need to show that P (0) is true i.e., 3 (0 3 0) Now (0 3 0) = 0 and since 0 = 0 × 3, 3 0. Hence P (0) is true Inductive step: Let k N such that P ( k ) is true i.e., 3 ( k 3 k ) [ IH ] We need to show that 3 (( k +1) 3 ( k +1)) Now (( k +1) 3 ( k
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am_lect_17 - Example Claim: n , n(n + 1) i = 2 i =1 n 0...

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