A bounded sequence of length n is a sequence of integers so that the next element is either one greater than or less than the previous element.

Here are a few examples of bounded sequences of length 8, starting with1: (1,2,3,4,3,2,1,0),(1,2,1,2,1,2,1,2),(1,0,1,2,3,4,3,2) 1. For m,n1, how many bounded sequences of length n are there that start with an integer between 1 and m ? (give a brief justification.) (5 points) 2. How many bits would the most efficient encoding of such sequences use? (in terms of n and m. Justify your work) (5 points) 3. Develop your own encoding/decoding algorithm where the code uses this number of bits. Describe your algorithms, and show the decoding recovers the encoded message. (10 points) 4. How many bounded sequences of length n have first and last element 0? (10 points) Here are a few examples of bounded sequences of length 8, starting with1: (1,2,3,4,3,2,1,0),(1,2,1,2,1,2,1,2),(1,0,1,2,3,4,3,2) 1. For m,n1, how many bounded sequences of length n are there that start with an integer between 1 and m ? (give a brief justification.) (5 points) 2. How many bits would the most efficient encoding of such sequences use? (in terms of n and m. Justify your work) (5 points) 3. Develop your own encoding/decoding algorithm where the code uses this number of bits. Describe your algorithms, and show the decoding recovers the encoded message. (10 points) 4. How many bounded sequences of length n have first and last element 0? (10 points)