Question

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

Answer 1

First, put the equation of the line given into slope-intercept form by solving for y. You get y = 2x +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2x + b and solving for b, we get b = 6