Use the binomial theorem to prove the following equalities for any natural number n ∈ n. it is enough in each case to specify x and y in the binomial theorem and show that this results in the given equality. (i’ve written the left-hand sides in two ways–in summation notation and written out in full.) (a) (hammack §3.4 #5 ) xn k=0 n k = n 0 + n 1 + · · · + n n = 2n (b) (hammack §3.4 #7 ) xn k=0 3 k n k = n 0 + 3 n 1 + 32 n 2 + · · · + 3n n n

14 pointslet x be the seats in the small van and y the seats in the large van. how many seats does the large van have? a company uses two vans to transport worker from a free parking lot to the workplace between 7: 00 and 9: 00a. m. one van has 9 more seats than the other. the smaller van makes two trips every morning while the larger one makes only one trip. the two vans can transport 69 people, maximum. how many seats does the larger van have?

Use the quadratic formula to solve the equation. if necessary, round to the nearest hundredth. x^2 - 23 = 10x a. -1.93, 11.93 b. 1.93, -11.93 c. 1.93, 11.93 d. -1.93, -11.93

How do the definitions, properties, and postulates you've learned about line segments, parallel lines, and angles to define the shapes of polygons