A jeweler had a fixed amount of gold to make bracelets and necklaces. The amount of gold in each bracelet is 7 grams and the amount of gold in each necklace is 20 grams. The jeweler used 109 grams of gold and made 4 more bracelets than necklaces. Determine the number of bracelets made and the number of necklaces made.

3 necklaces and 12 bracelets

Step-by-step explanation:

Bracelets = x +4

amount of necklace gold  =   20*x

amount of gold in bracelet = 7*(x+4)    

added together = 109 gm

 20x    +  7(x+4)     =    109    

20x + 7x+28 = 109

27x+28=109

-28        -28

27x= 81

/27   /27

x=3 necklaces

then bracelets = 3+4=12

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