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rec0712_key - COT3100 Summer2001 Recitation on...

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COT3100 Summer’2001 Recitation on induction (Solutions) 07/12/2001 1. Use mathematical induction to prove that 3 divides n 3 +2 n whenever n is a nonnegative integer. , We are going to prove by induction on n 0, that n 3 +2 n is divisible by 3. Basis . n =0. 0 3 +0=0 is divisible by 3, so the proposition holds for n =0. IH . Assume that the proposition is true for n = k , where k is some integer, k 0. In other words we assume that k 3 +2 k is divisible by 3, i.e. k 3 +2 k = 3 p , p is an integer. IS . We need to prove that ( k +1) 3 +2( k +1) is divisible by 3. ( k +1) 3 +2( k +1) = k 3 +3 k 2 +3 k +1+2 k +2= = ( k 3 +2 k ) +3( k 2 + k +1) = 3 p +3 ( k 2 + k +1) by IH Since both terms have factor 3, the sum is divisible by 3. So, induction step is proved. By Induction Principle the proposition is true for any n . 2. Use induction to prove that 3+3 5+3 5 2 +… +3 5 n = 3(5 n +1 - 1)/4 We are going to prove by induction on n 0 that the equality is true. Basis . n =0. LHS =3, RHS=3(5 0+1 - 1)/4=3. So, LHS=RHS. IH . Assume that the proposition is true for n = k , where k is some integer k 0. In other words we assume that for some k 0 3+3 5+3 5 2 +… +3 5 k = 3(5 k +1 - 1)/4 IS . We need to prove that 3+3
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rec0712_key - COT3100 Summer2001 Recitation on...

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