Question 1 A)Given the recursive formula for the arithmetic sequence a1=−5, an=an−1+4, find the first three terms of the sequence.(2 points) a1=$$, a2=$$, and a3=$$ Question 2 A) Write out the first five terms of the sequence an=(−2)n. Determine if the sequence is arithmetic or geometric, and then find the common ratio or difference. Which option below is accurate? Option #1: The sequence is arithmetic, and the common difference is 6. Option #2: The sequence is arithmetic, and the common difference is −2. Option #3: The sequence is geometric, and the common ratio is −2. Option #4: The sequence is geometric, and the common ratio is 2. (1 point) Option #$$ is accurate. Question 3 A)What is the common difference in the recursively defined arithmetic sequence a1=4, an=an−1−5.(1 point) The common difference is $$. Question 4 A) Which option is the correct recursive equation for the arithmetic sequence 6, 2, −2, −6, . . .? Option #1: a1=6, an=an−1+4 Option #2: a1=6, an=an−1−4 Option #3: a1=6, an=4an−1 Option #4: a1=6, an=−4an−1 (1 point) Option #$$ is the correct recursive formula for the sequence. Question 5 A)Find the 7th value in the arithmetic sequence described by the explicit formula An=2.5+(n−1)4.(1 point) A7=$$

Question

GPT-4o mini · Answer

Let's solve the questions step by step:

Question 1

Given:

$ a_1 = -5 $
$ a_n = a_{n-1} + 4 $

Finding the first three terms:

$ a_1 = -5 $
$ a_2 = a_1 + 4 = -5 + 4 = -1 $
$ a_3 = a_2 + 4 = -1 + 4 = 3 $

Thus,

$ a_1 = -5 $
$ a_2 = -1 $
$ a_3 = 3 $

Question 2

Sequence: $ a_n = (-2)^n $

Finding the first five terms:

$ a_1 = (-2)^1 = -2 $
$ a_2 = (-2)^2 = 4 $
$ a_3 = (-2)^3 = -8 $
$ a_4 = (-2)^4 = 16 $
$ a_5 = (-2)^5 = -32 $

The first five terms are: -2, 4, -8, 16, -32.

Determining if the sequence is arithmetic or geometric:

Arithmetic: The common difference is calculated as $ a_n - a_{n-1} $.
- $ 4 - (-2) = 6 $
- $ -8 - 4 = -12 $
- $ 16 - (-8) = 24 $
- $ -32 - 16 = -48 $
The differences are not constant, hence not arithmetic.
Geometric: The common ratio is calculated as $ \frac{a_n}{a_{n-1}} $.
- $ \frac{4}{-2} = -2 $
- $ \frac{-8}{4} = -2 $
- $ \frac{16}{-8} = -2 $
- $ \frac{-32}{16} = -2 $
The common ratio is constant and equals -2.

Thus, the sequence is geometric, with the common ratio of -2.

Correct option:

Option #3: The sequence is geometric, and the common ratio is −2.

Question 3

Given:

$ a_1 = 4 $
$ a_n = a_{n-1} - 5 $

Finding the common difference: The common difference in an arithmetic sequence defined recursively as $ a_n = a_{n-1} + d $ is simply the value being added, which is -5 in this case.

Thus,

The common difference is -5.

Question 4

Finding the correct recursive equation for the sequence 6, 2, −2, −6, . . .:

The sequence decreases from 6 to 2, then from 2 to -2, and from -2 to -6.

Calculating the differences:

$ 2 - 6 = -4 $
$ -2 - 2 = -4 $
$ -6 - (-2) = -4 $

This shows that the common difference is -4.

Checking the options:

Option #1: $ a_1 = 6, a_n = a_{n-1} + 4 $ (not correct - incorrect difference)
Option #2: $ a_1 = 6, a_n = a_{n-1} - 4 $ (correct)
Option #3: $ a_1 = 6, a_n = 4a_{n-1} $ (not correct)
Option #4: $ a_1 = 6, a_n = -4a_{n-1} $ (not correct)

Thus, the correct recursive formula is Option #2.

Question 5

Explicit formula: $ A_n = 2.5 + (n-1) \cdot 4 $

Finding $ A_7 $: \[ A_7 = 2.5 + (7-1) \cdot 4 \] \[ = 2.5 + 6 \cdot 4 \] \[ = 2.5 + 24 \] \[ = 26.5 \]

Thus,

$ A_7 = 26.5 $

Summary of Answers:

$ a_1 = -5, a_2 = -1, a_3 = 3 $
Option #3 is accurate.
The common difference is -5.
Option #2 is the correct recursive formula for the sequence.
$ A_7 = 26.5 $