Assignment 2 - Written component Math 100 - Fall 2024 Due: Friday, September 27 at 11:00 pm Please complete each problem on a separate piece of paper. You will need to scan your work and submit it electronically through Assign2. (1) [12 pts] Use induction to show that 5"+1 - 4n + 11 is divisible by 16 for all n 2 1. (2) [12 pts] Use induction to prove that sin x + sin 3x + sin 5x + . . . + sin(2n - 1)x = 1 - cos 2nx 2 sin x for all n 2 1. (Hint: You may need the identity 2 sin A sin B = cos(A-B) -cos(A + B).) (3) (a) [7 pts] Find the limit lim (e - 373 - 2n2) n-+ 00 (b) [7 pts] What is the value of c for which the following limit is finite and non-zero? For this c, find the value of the limit. cn lim n too 22n+ 2 + (-1)n 4n+1 + 2n' (4) [20 pts] A sequence {an}n=1 is given by a1 = 1 and anti = Van + 20, for n 2 2. (a) [14 pts] Use mathematical induction to prove that the sequence {an }=, is increas ing and bounded (1 1). (b) [6 pts] Is {an}ne, convergent? Explain. If {an }ne, is convergent, find lim an. n -+ 0o