Question

1. If you know the fifth term in an arithmetic sequence and the common difference, can you write a recursive formula for the sequence?

a. Yes
b. No
*Explanation:* In an arithmetic sequence, the recursive formula is given by ( a_n = a_n-1 + d ), where ( a_n ) is the nth term, ( a_n-1 ) is the (n-1)th term, and ( d ) is the common difference.

2. In your own words, tell how geometric sequences are related to exponential functions. Share your answer with the rest of the group.
a. Geometric sequences are not related to exponential functions.
b. Geometric sequences and exponential functions share similarities.
c. Geometric sequences and exponential functions are entirely different concepts.
d. None of the above.
*Explanation:* Geometric sequences are related to exponential functions because the terms in a geometric sequence can be expressed as exponential functions, such as ( a_n = a_1 times r^(n-1) ).

3. When a biological researcher arrived on Wombat Island, there were about 1000 wombats on the island. The researcher modeled the population growth with the explicit formula ( a_n = 1000 times 2^n - 1 ). Do you agree with the researcher's method of predicting the populations for years 6, 10, and 50?
a. Yes
b. No

Answer 1

Yes, you can write a recursive formula for an arithmetic sequence given the fifth term and the common difference. Geometric sequences are related to exponential functions. The reliability of the researcher's method of predicting the populations for years 6, 10, and 50 is uncertain without more information.

Step-by-step explanation:

In an arithmetic sequence, the recursive formula is given by an = an-1 + d, where an is the nth term, an-1 is the (n-1)th term, and d is the common difference. So, if you know the fifth term in an arithmetic sequence and the common difference, you can write a recursive formula for the sequence.

Geometric sequences are related to exponential functions because the terms in a geometric sequence can be expressed as exponential functions, such as an = a1 * r(n-1).

Regarding the researcher's method of predicting the populations for years 6, 10, and 50 using the explicit formula an = 1000 * 2n-1, it would depend on whether the model accurately represents the actual population growth of wombats on Wombat Island. Without more information, it is difficult to determine if the researcher's predictions are reliable.