Strong Induction (11 points) (1) (6 points) Let P(n) be the statement that a postage of n cents can be formed using just 3-cent stamps and 7-cent stamps. The Induction and Recursion parts of this exercise outline a strong induction proof that P(n) is true for n 18. (a) (1 points) Show that P(12), P(13), and P(14) are true, which com- pletes the base case. (b) (1 points) What is the inductive hypothesis? (c) (1 points) What do you need to prove in the inductive step? (d) (3 points) Complete the inductive step for k 14. (2) (5 points) Use strong induction to show that every positive integer can be written as a sum of distinct powers of two (i. e., 20 = 1; 21 = 2; 22 = 4; 23 = 8; 24 = 16; : : :). For example: 19 = 16 + 2 + 1 = 24 + 21 + 20 Hint: For the inductive step, separately consider the case where k +1 is even and where it is odd. When it is even, note that (k + 1)=2 is an integer.

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

7 ft 6 in means 7 1/2, because 6 in is 1/2 of a foot. 7 ft 8 in means 7 2/3 because 8 in is 2/3 of a foot. 23/3 = 7 2/3. 15/2 = 7 1/2.

23/3   *   15/2   = 57 1/2

step-by-step explanation: