Let p and q be the propositions p: I bought a lottery ticket this week. q :I won the million dollar jackpot. Express each of these propositions as an English sentence.
a)¬p
b) p∨q
c) p → q
d) p∧q
e) p ↔ q
f) ¬p →¬ q
g) ¬p∧¬q
h) ¬p∨(p∧ q)

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt

step-by-step explanation:

answer: 12x^2+6x

step-by-step explanation:

6x does not equal 24x. hope it works!