Given the recursive formula for the arithmetic sequence a1=−5

Question

, an=an−1+4
, find the first three terms of the sequence.(2 points)
a1=
, a2=
, and a3=

Answer 1

To find the first three terms of the arithmetic sequence defined by the given recursive formula, we can calculate \(a_2\) and \(a_3\) using the initial term \(a_1\).

We are given:

• \(a_1 = -5\)
• The recursive formula \(a_n = a_{n-1} + 4\).

Now, let's calculate \(a_2\) and \(a_3\):

1. Calculate \(a_2\): \[ a_2 = a_1 + 4 = -5 + 4 = -1 \]

2. Calculate \(a_3\): \[ a_3 = a_2 + 4 = -1 + 4 = 3 \]

Thus, the first three terms of the sequence are:

• \(a_1 = -5\)
• \(a_2 = -1\)
• \(a_3 = 3\)

So, the results are:

• \(a_1 = -5\)
• \(a_2 = -1\)
• \(a_3 = 3\)

Answer 2

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.
Remaining Attempts : 1
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Answer 3

To determine if the sequence defined by \( a_n = (-2)^n \) is arithmetic or geometric, we first need to calculate the first five terms of the sequence.

1. Calculate the first five terms:

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

   Therefore, the first five terms are:

   • \( a_1 = -2 \)
   • \( a_2 = 4 \)
   • \( a_3 = -8 \)
   • \( a_4 = 16 \)
   • \( a_5 = -32 \)

2. Determine if the sequence is arithmetic or geometric:

   • Arithmetic Sequence: A sequence is arithmetic if the difference between consecutive terms is constant.

     • \( a_2 - a_1 = 4 - (-2) = 4 + 2 = 6 \)
     • \( a_3 - a_2 = -8 - 4 = -12 \)
     • \( a_4 - a_3 = 16 - (-8) = 16 + 8 = 24 \)
     • \( a_5 - a_4 = -32 - 16 = -48 \)

     The differences are not constant, so the sequence is not arithmetic.

   • Geometric Sequence: A sequence is geometric if the ratio between consecutive terms is constant.

     • \( \frac{a_2}{a_1} = \frac{4}{-2} = -2 \)
     • \( \frac{a_3}{a_2} = \frac{-8}{4} = -2 \)
     • \( \frac{a_4}{a_3} = \frac{16}{-8} = -2 \)
     • \( \frac{a_5}{a_4} = \frac{-32}{16} = -2 \)

     The ratios are constant, specifically \(-2\).

3. Conclusion: Since the sequence is geometric, and the common ratio is \(-2\).

The correct option is: Option #3: The sequence is geometric, and the common ratio is -2.

Answer 4

What is the common difference in the recursively defined arithmetic sequence a1=4
, an=an−1−5
.(1 point)
The common difference is
.

Answer 5

In the recursively defined arithmetic sequence given by:

• \( a_1 = 4 \)
• \( a_n = a_{n-1} - 5 \)

the common difference can be found from the recursive formula. In this case, the common difference is indicated by the term that is subtracted from each term to obtain the next term in the sequence.

Since the recursive formula states that each term is the previous term minus 5, the common difference \(d\) is:

\[ d = -5 \]

Therefore, the common difference in the sequence is \(-5\).

Answer 6

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.
Remaining Attempts : 1

Answer 7

To determine the correct recursive equation for the arithmetic sequence \(6, 2, -2, -6, \ldots\), we need to identify the common difference in the sequence.

1. Calculate the common difference:
   • \(2 - 6 = -4\)
   • \(-2 - 2 = -4\)
   • \(-6 - (-2) = -4\)

The common difference (\(d\)) is \(-4\).

2. Write the recursive definition:
   • The first term \(a_1 = 6\)
   • The recursive equation that captures the pattern in this sequence is:
     \[ a_n = a_{n-1} - 4 \]

Now let’s review the options provided:

• Option #1: \( a_1 = 6, a_n = a_{n-1} + 4 \) (Not correct)
• 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)

The correct option is: Option #2: \( a_1 = 6, a_n = a_{n-1} - 4 \).

Answer 8

Find the 7th value in the arithmetic sequence described by the explicit formula An=2.5+(n−1)4
.(1 point)
A7=

Answer 9

To find the 7th value in the arithmetic sequence described by the explicit formula:

\[ A_n = 2.5 + (n - 1) \cdot 4 \]

we can substitute \(n = 7\) into the formula:

\[ A_7 = 2.5 + (7 - 1) \cdot 4 \]

Now, calculate the term step-by-step:

1. Calculate \(7 - 1\): \[ 7 - 1 = 6 \]

2. Multiply by \(4\): \[ 6 \cdot 4 = 24 \]

3. Add \(2.5\): \[ A_7 = 2.5 + 24 = 26.5 \]

Thus, the 7th value in the sequence is:

\[ A_7 = 26.5 \]