I don't really know where to start with this. I'm not looking for just an answer but an explanations of how it's done.

Block Conversion from Binary to Octal We have learned that we can block-convert a binary number to octal by grouping the binary number into blocks of 3 digits (going from right to left, perhaps padding the binary number with one or two leading zeros to complete the leftmost block of three) and converting each 3 digit binary number into one octal digit. The following is a proof that this procedure works as advertised. Suppose n is a nonnegative integer and its binary expansion is given by m n = Z dkzk k=0 where each dk 6 {0,1} and m is a nonnegative integer. We can assume without loss of generality that the number of terms in this sum is a multiple of 3,i.e. m + 1 = 34:; for some natural number q. We now group the sum into blocks of 3 terms each, as follows: 0-1 2 Q-1 2 0-1 Tl. = Z d3i+j23+j :2? 20 d3i+j2j = QZ 8i(d3i + 2d3i+1 + 4d3i+2) i=0 j=0 i=0 =0 i=0 We now set 01- = dgi + 2d35+1 + 4d3i+2 for all i and get q-1 n = Z Gil-3i i=0 Since each dk is 0 or 1, the of satisfy 0 S o,- S 7, i.e., there are octal digits. We have found the octal expansion of n, and it is obtained by block-converting three binary digits at a time to octal, from right to left. Questions: 1. 9. What is the meaning of \"without loss of generality\"? What is the purpose here of that device? How can the number of terms in the sum, an arbitrary integer, always be a multiple of 3? [V] What does the quantity q represent? {V} Why is the single sigma sum equal to the double sigma sum? [VA] What algebraic laws are used to simplify the sigma sum? Why could the 23' be factored out of the inner sigma sum, but not the outer one? [V] Fill in a detail that the given proof omits, namely why the o,- satisfy the inequality 0 S o,- 5 7. [VA] Rewrite the given proof to demonstrate how and why block conversion from octal to binary works. [P] Write a similar proof to demonstrate how and why block conversion from binary to hexadecimal works. [P] You can't block convert from binary to decimal. At which specic point(s] would a proof that follows the lines of the proof given here fail? [V] For which numbers I) can binary numbers be block converted to base-b? [Answer only, no explanation necessary] 10. Write a proof that block conversion aiways works for those numbers. [P] Questions marked [P] require a rigorous proof. Questions marked [V] 'onlif require a clear, succinct verbal explanation. [VA] requires such a verbal explanation, backed up by algebra, but omitting the formalities of a rigorous proof, such as strict, explicit attention to quantification, assumptions and conclusions. I may refuse to even read explanations that are lengthy, meandering, vague, or all three