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Define P(n) to be the assertion that: n n j= j=1 = (a) Verify that P(3) is true. (b) Express P(k). (c) Express P(k + 1). (d) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6 n(n + 1) (2n + 1) 6 what must be proven in the base case? (e) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6

Define P(n) to be the assertion that: n n j= j=1 = (a) Verify that P(3) is true. (b) Express P(k). (c) Express P(k + 1). (d) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6 n(n + 1) (2n + 1) 6 what must be proven in the base case? (e) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 42E
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Define P(n) to be the assertion that: = n (a) Verify that P(3) is true. (b) Express P(k). (c) Express P(k+ 1). (d) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6 n n Σ₁². j=1 what must be proven in the base case? = j=1 = (e) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6 n Σj² = j=1 n(n + 1) (2n + 1) 6 = what must be proven in the inductive step? (f) What would be the inductive hypothesis in the inductive step from your previous answer? (g) Prove by induction that for any positive integer n, n(n + 1)(2n + 1) 6
Transcribed Image Text:Define P(n) to be the assertion that: = n (a) Verify that P(3) is true. (b) Express P(k). (c) Express P(k+ 1). (d) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6 n n Σ₁². j=1 what must be proven in the base case? = j=1 = (e) In an inductive proof that for every positive integer n, n(n + 1) (2n + 1) 6 n Σj² = j=1 n(n + 1) (2n + 1) 6 = what must be proven in the inductive step? (f) What would be the inductive hypothesis in the inductive step from your previous answer? (g) Prove by induction that for any positive integer n, n(n + 1)(2n + 1) 6
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