Prove the theorem below using the techniques ofbinding the term and splitting thesumorapproximation by integralsto find a tight bound for the sum. Make sure yourproof is complete, concise, clear and precise. Theorem 4.n∑i=1id∈Θ(nd+1)(5 points) 2.Prove the theorem below using the techniques ofbinding the term and splitting thesumorapproximation by integralsto find a tight bound for the sum. Make sure yourproof is complete, concise, clear and precise. Theorem 5.n∑i=1(log2i)c∈Θ(n(log2n)c)(10 points) 3. Consider the recurrenceT(n).T(n)={cifn≤12T(⌊n4⌋) +16ifn>1a)Use the recursion tree or repeated substitution method to come up with a goodguess for a boundf(n) on the recurrenceT(n).b) State and prove by induction two theorems showingT(n)∈Θ(f(n)).GradingYou will be docked points for errors in your math, disorganization, unclarity, orincomplete proofs

Determine whether the quadrilateral below is a parallelogram. justify/explain your answer (this means back it up! give specific information that supports your decision. writing just "yes" or "no" will result in no credit.)

Precalculus question, image attached.

In a test for esp (extrasensory perception), the experimenter looks at cards that are hidden from the subject. each card contains either a star, a circle, a wave, a cross or a square.(five shapes) as the experimenter looks at each of 20 cards in turn, the subject names the shape on the card. when the esp study described above discovers a subject whose performance appears to be better than guessing, the study continues at greater length. the experimenter looks at many cards bearing one of five shapes (star, square, circle, wave, and cross) in an order determined by random numbers. the subject cannot see the experimenter as he looks at each card in turn, in order to avoid any possible nonverbal clues. the answers of a subject who does not have esp should be independent observations, each with probability 1/5 of success. we record 1000 attempts. which of the following assumptions must be met in order to solve this problem? it's reasonable to assume normality 0.8(1000), 0.2(1000)%30 approximately normal 0.8(1000), 0.2(1000)% 10 approximately normal srs it is reasonable to assume the total number of cards is over 10,000 it is reasonable to assume the total number of cards is over 1000