chap mathematical preliminaries (bgive the summation that results from expanding your recurrence (cgive closed-form solution for the summation prove (using inductionthat the recurrence (nt( nt( has as its closed-form solution (nn( )/ expand the following recurrence to help you find closed-form solutionand then use induction to prove your answer is correct ( ( for ( expand the following recurrence to help you find closed-form solutionand then use induction to prove your answer is correct (nt( for ( assume that an -bit integer (represented by standard binary notationtakes any value in the range to with equal probability (afor each bit positionwhat is the probability of its value being and what is the probability of its value being (bwhat is the average number of " bits for an -bit random number(cwhat is the expected value for the position of the leftmost " bitin other wordshow many positions on average must we examine when moving from left to right before encountering " bitshow the appropriate summation what is the total volume of your body in liters (orif you prefergallons) an art historian has database of , full-screen color images (aabout how much space will this requirehow many cd-roms would be required to store the database( cd-rom holds about mb of databe sure to explain all assumptions you made to derive your answer (bnowassume that you have access to good image compression technique that can store the images in only / of the space required for an uncompressed image will the entire database fit onto single cdrom if the images are compressed how many cubic miles of water flow out of the mouth of the mississippi river each daydo not look up the answer or any supplemental facts be sure to describe all assumptions made in arriving at your answer