MATH Homework

MATH Homework
Wenjian Liu (Instructor) MA-471 | Fall 2022 Homework 7Due Sunday, November 27, 2022Problem 7.1 Prove that n X j =1 j 4 = n (n + 1)(2 n+ 1) 3 n 2 + 3 n 1 30 whenever nis a positive integer. Problem 7.2 Prove that for every positive integer n, 1 2 + 2 3 + +n(n + 1) = n (n + 1)( n+ 2) 3 Problem 7.3 Prove that for every positive integer n, 1 + 1 p 2 + 1 p 3 + + 1 p n > 2(p n + 1 1) Problem 7.4 Use mathematical induction to prove that 5 divides n5 n whenever nis a nonnegative integer. Problem 7.5 Prove that if nis a positive integer, then 133 divides 11n +1 + 12 2 n 1 Problem 7.6 Prove, for all positive integers n, 1 2 3 4 2 n 1 2 n < 1 p 3 n Problem 7.7 Denote by f n the nth Fibonacci number. Show that when nis a positive integer, f 0f 1 + f 1f 2 + +f 2n 1f 2n = f2 2 n 1 Wenjian Liu (Instructor) MA-471 | Fall 2022 Homework 7Due Sunday, November 27, 2022Problem 7.8 The sequence fa ng satis es the recurrence relation an = 8 a n 1 + 10 n 1 and the initial condition a 1 = 9. Use generating functions to nd an explicit formula for a n. 2