Joslyn is planting daisy seeds in a flower garden. Each day, she plants a different number of seeds. The list below shows the number of seeds Joslyn planted on each of the first six days.

Question

Joslyn is planting daisy seeds in a flower garden. Each day, she plants a different number of seeds. The list below shows the number of seeds Joslyn planted on each of the first six days.
1, 2, 4, 8, 16, 32

Joslyn wants to determine the number of daisy seeds, d, she should plant on the nth day. Which statements below are true? Select two that apply.

A. 
The sequence is arithmetic.

B. 
The sequence is geometric.

C. 
The sequence is represented by d = 2n.

D. 
The sequence is represented by d= 2n - 1.

E. 
The sequence is represented by d = 2^n .

F. 
The sequence is represented by d = 2^n-1 .

GPT-4o mini · Answer

To determine which statements are true regarding the sequence of seeds Joslyn plants, we can analyze it step by step.

The number of seeds planted each day is:
Day 1: 1 (which is $2^0$)
Day 2: 2 (which is $2^1$)
Day 3: 4 (which is $2^2$)
Day 4: 8 (which is $2^3$)
Day 5: 16 (which is $2^4$)
Day 6: 32 (which is $2^5$)

From this, we can see that the number of seeds doubles each day. Therefore, the properties of the sequence can be evaluated as follows:

A. The sequence is arithmetic.
This is FALSE. An arithmetic sequence has a common difference, which is not the case here since the number doubles (the ratios are constant, not the differences).

B. The sequence is geometric.
This is TRUE. A geometric sequence has a common ratio. The ratio here is 2 (each term is multiplied by 2 to get to the next term).

C. The sequence is represented by d = 2n.
This is FALSE. The formula does not hold for the values given.

D. The sequence is represented by d = 2n - 1.
This is FALSE. This formula does not represent the correct sequence either.

E. The sequence is represented by d = 2^n.
This is TRUE, but it doesn’t match perfectly with the days. For example, on Day 1, using $d = 2^n$ gives 2 seeds (when $n = 1$), which does not match the number of seeds actually planted, which corresponds to Day numbers starting from $n = 0$ for $d = 2^n$. Thus, this can also be rewritten as (d = 2^{n-1}$ for $n = 1, 2, 3,$ etc.

F. The sequence is represented by d = 2^n - 1.
This is FALSE. This expression would imply that the number of seeds starts from 1 and gives incorrect values as well.

The two true statements, therefore, are:

B. The sequence is geometric.
E. The sequence is represented by d = 2^n (though keep in mind it starts from $d = 2^0$ for Day 1).

If we interpret $E$ to mean $d = 2^{n-1}$, which reflects the actual planting days, it's indeed valid, but as it stands strictly, it is considered not starting from Day 1 correctly. Hence the best interpretations in this formulation lead us back to agreeing with B as being clearly true, while E can be further justified contextually depending on the formulation of the seeds planted.