Geoff planted dahlias in his garden. Dahlias have bulbs that divide and reproduce underground. In the first year, Geoff's garden produced 8 bulbs. In the second year, it produced 16 bulbs, and in the third year it produced 32 bulbs. If this pattern continues, how many bulbs should Geoff expect in the sixth year?
Geoff should expect 128 bulbs in the sixth year.
To see why, notice that each year the number of dahlias doubles. So after 3 years, there are 2^3 = 8 times as many dahlias as there were in the first year. After 4 years, there are 2^4 = 16 times as many, and after 5 years there are 2^5 = 32 times as many.
Therefore, in the sixth year, there should be 2^6 = 64 times as many dahlias as in the first year. Since there were originally 8 bulbs, we can find the number of bulbs in the sixth year by multiplying 8 by 64:
8 * 64 = 512
But remember, each bulb can divide and produce new bulbs, so this is just an estimate. The actual number of bulbs in the garden may be higher or lower depending on various factors.