Geoff planted dahlias in his garden. daliahs have bulbs that divide and reproduce underground. in the first year, geoff's garden produced 6 bulbs. in the second year, it produced 12 bulbs, and in the third year, it produced 24 bulbs. if this pattern continues, how many bulbs should geoff expect in the eighth year?
a)48 bulbs
b)384 bulbs
c)768 bulbs
d)1,536
i think it's b?

This is a geometric sequence of the form:

a(n)=ar^(n-1), a=initial term, r=common ratio, n=term number

The common ratio, r, is the defining characteristic of geometric sequences.  It is the value found when dividing each term by the previous term.  In this case:

r=24/12=12/6=2 and we see that the initial value is 6 so

a(n)=6(2)^(n-1) so in the eighth year:

a(8)=6(2^7)

a(8)=768

So after eight years there will be 768 bulbs.

First Rectangle: P= 14; L= 4, W= ?
Second Rectangle: P= ?, L= ?, W= 9

STEP 1:
solve for the width in the first rectangle using perimeter formula

Perimeter= 2L + 2W
14= 2(4) + 2W

14= 8 + 2W
subtract 8 from both sides

6= 2W
divide both sides by 2

3= W

STEP 2:
Since the two rectangles are similar, they may be different sizes, but they have corresponding angles. We can set up a ratio to compare width and length.

First Rectangle: P= 14, L= 4, W= 3
Second Rectangle: L= ?, W= 9

length/width= length/width
4/3= L/9

cross multiply
4 * 9= 3 * L

36= 3L
divide both sides by 3

12= L

STEP 3:
find perimeter of second rectangle; use length from step 2

Perimeter= 2L + 2W
=2(12) + 2(9)
=24 + 18
=42 feet

ANSWER: (C) 42 feet

Hope this helps! :)