Question

Write a recursive function for each sequence.81,85,89,93,97

Answer 1

The recursive function for the arithmetic sequence 81, 85, 89, 93, 97 is defined by the formula a_n = a_{n-1} + 4, for n > 1, with the initial condition a_1 = 81. The sequence has a common difference of 4.

Step-by-step explanation:

To write a recursive function for the given sequence 81, 85, 89, 93, 97, we must first identify the pattern. The sequence increases by 4 each time, which is a common difference in arithmetic sequences. With this information, we can establish the recursive formula.

The recursive formula for an arithmetic sequence is given by:

a_n = a_{n-1} + d

where:

• a_n is the nth term of the sequence,
• a_{n-1} is the previous term,
• d is the common difference between the terms (which in this case is 4).

For the initial condition:

a_1 = 81

Therefore, the recursive function for the given sequence is:

a_n = a_{n-1} + 4, for n > 1, with a_1 = 81

Answer 2

Okay, here we have this:

Considering the provided secuence: 81,85,89,93,97. Let's calculate the difference between each term to identify if it is constant or not:

85-81=4

89-85=4

93-89=4

97-93=4

Here we can see that the difference is a constant, so this mean that it's an Arithmetic sequence, now, let's replace in the formula:

`a_n=a_1+(n-1)d`

Replacing:

`a_n=81+(n-1)4`

And, turning this in a recursive function:

`g(n)=\begin{cases}81\text{ if n=}1 \\ g(n-1)+4\text{ if }n>1\end{cases}`