Consider the sequence 41,29,17,5

Question

The recursive Function A1__ with

=_____.

A
n
=
A
n
−
1
−
12
A
n
​
=A
n−1
​
−12
, 41

A
n
=
12
A
n
−
1
A
n
​
=12A
n−1
​

,
−
12
−12

A
n
=
A
n
−
1
+
12
A
n
​
=A
n−1
​
+12
, 53

Answer 1

To find the correct recursive function for the sequence \( 41, 29, 17, 5 \), let's first observe the pattern in the sequence.

1. Initial terms:

   • \( A_1 = 41 \)
   • \( A_2 = 29 \)
   • \( A_3 = 17 \)
   • \( A_4 = 5 \)

2. Differences:

   • \( A_2 - A_1 = 29 - 41 = -12 \)
   • \( A_3 - A_2 = 17 - 29 = -12 \)
   • \( A_4 - A_3 = 5 - 17 = -12 \)

From the above calculations, we see that each term in the sequence is obtained by subtracting 12 from the previous term.

This leads to the following recursive formula: \[ A_n = A_{n-1} - 12 \quad \text{for } n \geq 2, \text{ with } A_1 = 41. \]

Thus, the answer is: \[ A_n = A_{n-1} - 12, \quad A_1 = 41. \]