In the first year Geoff's garden produced 4 bulbs, in the second year it produced 8, and the third 16. How many bulbs will geoff's garden produce on the 10th year?
A: 2,048
B: 1,024
C: 40
D: 32
We notice that the number of bulbs produced each year is a power of 2, specifically $2^0=1$ in the first year, $2^1=2$ in the second year, and $2^2=4$ in the third year.
If we continue this pattern, we see that the number of bulbs produced in the $n$th year is $2^{n-3}$. Therefore, the number of bulbs produced in the 10th year is $2^{10-3}=2^7=\boxed{\textbf{(D) }32}$.
To find the pattern in the number of bulbs produced by Geoff's garden, we can observe that the number of bulbs doubles each year.
In the first year, 4 bulbs were produced
In the second year, 8 bulbs were produced (double of the previous year)
In the third year, 16 bulbs were produced (double of the previous year)
We can see that the number of bulbs produced follows a geometric sequence, where the common ratio is 2.
To find out how many bulbs will be produced in the 10th year, we can use the formula for the nth term of a geometric sequence:
An = A1 * r^(n-1)
Where:
An is the nth term
A1 is the first term
r is the common ratio
n is the number of terms
Using the given information:
A1 = 4 (number of bulbs produced in the first year)
r = 2 (common ratio)
n = 10 (desired year)
Plugging in the values into the formula, we can find the number of bulbs produced in the 10th year:
A10 = 4 * 2^(10-1)
A10 = 4 * 2^9
A10 = 4 * 512
A10 = 2,048
Therefore, Geoff's garden will produce 2,048 bulbs in the 10th year.
The answer is option A: 2,048.