Question

Consider the sequence
The recursive Function is ___41,29,17,5____ with A_1

=_____.



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
, 41

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

Answer 1

To determine the recursive function and the first term \( A_1 \) of the sequence \( 41, 29, 17, 5 \), we can analyze the differences between consecutive terms.

- The difference between 41 and 29 is \( 41 - 29 = 12 \)
- The difference between 29 and 17 is \( 29 - 17 = 12 \)
- The difference between 17 and 5 is \( 17 - 5 = 12 \)

It seems that each term in the sequence decreases by 12. This suggests that the recursive relation is:

\[
A_n = A_{n-1} - 12
\]

Now, let's find the first term \( A_1 \).

From the given sequence:

- For \( n=4 \): \( A_4 = 5 \)
- For \( n=3 \): \( A_3 = A_4 + 12 = 5 + 12 = 17 \)
- For \( n=2 \): \( A_2 = A_3 + 12 = 17 + 12 = 29 \)
- For \( n=1 \): \( A_1 = A_2 + 12 = 29 + 12 = 41 \)

Based on this analysis:

The recursive function is:
\[
A_n = A_{n-1} - 12
\]

with \( A_1 = 41 \).