And show all work use mathematical induction to prove the statement is true for all positive integers n, or show why it is false. (4 points each.) 4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + + 4n( 4n + 2) = quantity four times quantity four n plus one times quantity eight n plus seven divided all divided by six 1^2 + 4^2 + 7^2 + + (3n - 2)^2 = quantity n times quantity six n squared minus three n minus one all divided by two for the given statement pn, write the statements p1, pk, and pk+1. (2 points) 2 + 4 + 6 + . . + 2n = n(n+1)

The inside wheels of a car traveling on a circular path are rotating half as fast as the outside wheels. the front two wheels are six feet apart. what is the number of feet in the path traced by the inside front wheel in one trip around the circle? express your answer in the form "k \pi", where k is an integer.

Graph the system of equations on your graph paper to answer the question. {y=−x+4y=x−2 what is the solution for the system of equations? enter your answer in the boxes.

Aconcession manager at yankee stadium wants to know how temperature affects beer sales. she took a sample of 10 games and recorded the number of beers sold and the temperature in the middle of the game. temperature 80 68 78 79 87 74 86 92 77 84 number of beers 20533 1439 13829 21286 30985 17187 30240 87596 9610 28742 a. draw a scatter plot of the data. b. the manager estimates the regression equation to be: numberofbeers = −100, 678 + 1, 513 ∗ temperature draw this on your scatter plot. c. for one of the estimated points, indicate the residual with ei . d. for that same point, indicate what part of the variation is explained by the model with ˆyi − y¯.