Madison made 1 & 1/2 gallons of punch she used 2 pints of punch 3 quarts of ginger ale the rest was orange juice how many quarts of orange juice did she use ?

2 quarts of orange juice

Step-by-step explanation:

If she made 1&1/2 gallons and, excluding orange juice, she used 2 pints (1quart) and 3 quarts that is 1 gallon (without orange juice) if we then subtract what we have from our total we are left with 1/2 gallon (aka: 2quarts) meaning we have a remaining 2 quarts of orange juice

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