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To prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Let P (n) be the statement that 12 + 22 +···+ n2 = n(n + 1)(2n + 1)/6 for the positive integer n.
a) What is the statement P (1)?
b) Show that P (1) is true, completing the basis step of the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step, identifying where you use the inductive hypothesis.
f ) Explain why these steps show that this formula is true whenever n is a positive integer
For the function f(x)=x-1-x2, determine if f'' is increasing or decreasing at x=2 and determine concavity at x=2. Find the distance from the origin to the point on the line y=-2x+5 that is closest to the origin.
As Jane's distance from home increases, how does her distance from work change? After Jane passes the mailbox, how long does it take her to travel 20 feet? 10 feet? 5 feet? Explain your thinking
Use Euler's method with step h=0.1 to find the first two approximations (y1 and y2) of the initial value problem y' = 3+x-y, y(0) = 3.
Coffee is draining - Application of calculus
find the diagonal distance across the baseball diamond from third base (T) to first base (F)
At the first tri-city meeting, there are 8 people from town A, 7 people from town B, and 5 people from town C. If a council consisting of 5 people is randomly selected, find the probability that 2 are from town A and 3 are from town B.
the perimeter of a basketball court is 276'. the length is 40' longer than the width. find the dimensions of the court.
A ball is thrown vertically upwards from a height of 5 ft with an initial velocity of 50 ftper second. How high will the ball go?
(a) Write an equation in x and y for the number of gallons of gas sold. (b) Write an equation in x and y for the amount of money received.
If a line y=3x+6 is tangent to the circle x^2+y^2=9 at point (k,k+1) then find value of k?
A health spa claims that a new exercise program will reduce a person's waist size by 2 centimeters on the average over a 5-day period. The waist sizes of 6 men who participated in this exercise program are recorded before and after the 5-day p..
Let v1 and v2 be vectors in R^n. Show that (prove) the set of all vectors w, an element of R^n such that w is orthogonal to both v1 and v2, is a subspace of R^n.
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