Final answer:
Yes, you can write a recursive formula for an arithmetic sequence if you know the fifth term and the common difference. Geometric sequences are related to exponential functions as both involve growth or decay through repeated multiplication by a constant ratio or base.
Step-by-step explanation:
Understanding Arithmetic Sequences and Exponential Functions:
If you know the fifth term in an arithmetic sequence and the common difference, can you write a recursive formula for the sequence? The answer is yes. Knowing the fifth term, let's say 'a5', and the common difference 'd', enables us to express the nth term recursively. Specifically, the recursive formula is given as:
an = an-1 + d for n > 1, with a1 = a5 - 4d, which provides us with the first term.
Next, geometric sequences are related to exponential functions due to their inherent nature of growth or decay by a constant ratio. Each term in a geometric sequence is found by multiplying the previous term by a fixed, non-zero number known as the common ratio (r). An exponential function, similarly, involves the repeated multiplication of a base raised to a variable exponent. In a geometric sequence, the positions (like time intervals) determine the exponent for the common ratio, closely resembling the exponential function's form. For instance, a sequence that doubles with each term resembles the exponential function 2n, where n represents the position or time interval.