Homework2.pdf - Seul-Ki Kim ICS 6D Homework 2 1 Exercise...

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Seul-Ki Kim ICS 6D January 23, 2019 Homework 2 1. Exercise 8.4.1 : Components of an inductive proof. (a-f) Define P(n) to be the assertion that: j ୨ୀଵ = ୬(୬ାଵ)(ଶ୬ା ) -------------------------------------------------------------------------------------------- (a) Verify that P(3) is true. 1 2 + 2 2 + 3 2 ? ଷ(ଷାଵ)(ଶ(ଷ)ାଵ) 1 + 4 + 9 ? ଷ(ସ)(଻) 14 = 14 (b) Express P(k). j ୨ୀଵ = ୩(୩ାଵ)(ଶ୩ାଵ) (c) Express P(k+1). j ୩ାଵ ୨ୀଵ = (୩ାଵ)(୩ାଵାଵ)(ଶ(୩ାଵ)ାଵ) = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) (d) In an inductive proof that for every positive integer n, j ୨ୀଵ = ୬(୬ାଵ)(ଶ୬ାଵ) What must be proven in the base case? For the base case, it has to be proven that formula is true for n = 1. When n = 1, the left side of the equation is j ୨ୀଵ = 1 . When n = 1, the right side of the equation is ଵ(ଵାଵ)(ଶାଵ) = 1 Therefore, j ୨ୀଵ = ଵ(ଵାଵ)(ଶାଵ) . (e) In an inductive proof that for every positive integer n, j ୨ୀଵ = ୬(୬ାଵ)(ଶ୬ାଵ) What must be proven in the inductive step? Suppose that for positive integer k, j ୨ୀଵ = ୩(୩ାଵ)(ଶ୩ାଵ) , then we will show that j = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) ୩ାଵ ୨ୀଵ Starting with the left side of the equation to be proven:
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j = ∑ j + (k + 1) ୨ୀଵ ୩ାଵ ୨ୀଵ by separating out the last term = ୩(୩ାଵ)(ଶ୩ାଵ) + (k + 1) by the inductive hypothesis = ୩(୩ାଵ)(ଶ୩ାଵ) + ଺(୩ାଵ) = (୩ାଵ)[୩(ଶ୩ାଵ)ା଺(୩ାଵ)] = (୩ାଵ)[ଶ୩ ା଻୩ା଺] = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) by algebra Therefore, j = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) ୩ାଵ ୨ୀଵ (f)
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