Prove these statements with induction, strong induction, or proof by the smallest counterexample: a. if n is an integer, then 1(2) + 2(3) + 3(4) + ... + n(n + 1) = [n(n + 1)(n + 2)]/3 b. Prove that the summation of (8i - 5) = 4n^2 - n for every positive integer n, where i = 1 to n. c. Prove that 3 | (5^[2n] - 1) for every integer n greater than or equal to 0. d. Prove that 9 | (4^[3n] + 8) for every integer n greater than or equal to 0. e. Prove that (1 + 2 + 3 + ... + n)^2 = 1^3 + 2^3 + 3^3 + ... + n^3 for every integer n.

Prove these statements with induction, strong induction, or proof by the smallest counterexample: a. if n is an integer, then 1(2) + 2(3) + 3(4) + ... + n(n + 1) = [n(n + 1)(n + 2)]/3 b. Prove that the summation of (8i - 5) = 4n^2 - n for every positive integer n, where i = 1 to n. c. Prove that 3 | (5^[2n] - 1) for every integer n greater than or equal to 0. d. Prove that 9 | (4^[3n] + 8) for every integer n greater than or equal to 0. e. Prove that (1 + 2 + 3 + ... + n)^2 = 1^3 + 2^3 + 3^3 + ... + n^3 for every integer n.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.4: Mathematical Induction

Problem 10E

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