Question

1. a recursive formula for the sequence 64, 48, 36, ... is

a. $a_n = 64(0.75)^{n - 1}$
b. $a_1 = 64$
$a_n = a_{n - 1} - 16$
c. $a_n = 64 + (n - 1)(-16)$
d. $a_1 = 64$
$a_n = 0.75a_{n - 1}$

Explanation:

Step1: Analyze the sequence type

First, check the ratio between consecutive terms: $\frac{48}{64} = 0.75$, $\frac{36}{48}=0.75$. So it's a geometric sequence? Wait, no, wait, 64 to 48: 48 = 64 0.75, 36 = 48 0.75. Wait, but let's check the options. Recursive formula for geometric sequence is $a_1 = \text{first term}$, $a_n = r \cdot a_{n - 1}$, where $r$ is the common ratio. Here, $r = 0.75$. Let's check each option:

Step2: Evaluate Option A

Option A is an explicit formula (since it's in terms of $n$ directly), not recursive. So A is out.

Step3: Evaluate Option B

Option B: $a_1 = 64$, $a_n = a_{n - 1} - 16$. Let's test: $a_2 = 64 - 16 = 48$? No, 64 - 16 is 48? Wait, 64 - 16 is 48, then $a_3 = 48 - 16 = 32$, but the sequence has 36. So B is wrong.

Step4: Evaluate Option C

Option C is an explicit formula (arithmetic sequence formula), not recursive. So C is out.

Step5: Evaluate Option D

Option D: $a_1 = 64$, $a_n = 0.75a_{n - 1}$. Test: $a_2 = 0.75 * 64 = 48$, $a_3 = 0.75 * 48 = 36$, which matches the sequence. So D is correct. Wait, but earlier I thought maybe arithmetic, but no, the ratio is constant (0.75), so it's geometric. So the recursive formula for geometric sequence is $a_1 = \text{first term}$, $a_n = r \cdot a_{n - 1}$. So D is correct. Wait, but wait, the user's sequence: 64, 48, 36,... Let's check the ratio again: 48/64 = 3/4 = 0.75, 36/48 = 3/4 = 0.75. So it's a geometric sequence with common ratio 0.75. So recursive formula is $a_1 = 64$, $a_n = 0.75a_{n - 1}$, which is option D. Wait, but earlier when I checked option B, I made a mistake: 64 - 16 is 48, but then 48 - 16 is 32, but the sequence has 36, so B is wrong. So D is correct.

Answer:

D. $a_1 = 64$, $a_n = 0.75a_{n - 1}$