Question

Which best describes the relationship between the successive terms in the sequence below?

–3.2, 4.8, –7.2, 10.8, …

The terms have a common difference of 1.6.
The terms have a common ratio of –1.5.
The terms have a common ratio of –0.67.
Answer
Which sequences are arithmetic? Check all that apply.

–5, 5, –5, 5, –5, …
96, 48, 24, 12, 6
18, 5.5, –7, –19.5, –32, …
–1, –3, –9, –27, –81, …
16, 32, 48, 64, 80
Answer
Which sequences are geometric? Check all that apply.

–2, –4, –6, –8, –10, …
16, –8, 4, –2, 1
–15, –18, –21.6, –25.92, –31.104, …
4, 10.5, 17, 23.5, 30, …
625, 125, 25, 5, 1, …
Answer
A sequence is defined by the recursive formula f (n + 1) = f(n) – 2. If f(1) = 18, what is f(5)?
Answer
yes 10!!




Which sequence could be partially defined by the recursive formula f (n + 1) = f(n) + 2.5 for n ≥ 1?

2.5, 6.25, 15.625, 39.0625, …
2.5, 5, 10, 20
–10, –7.5, –5, –2.5, …
–10, –25, 62.5, 156.25
Answer
Which recursive formula can be used to generate the sequence below, where f(1) = 3 and n ≥ 1?

3, –6, 12, –24, 48, …

f (n + 1) = –3 f(n )
f (n + 1) = 3 f(n )
f (n + 1) = –2 f(n )
f (n + 1) = 2 f(n)
Answered by GPT-4o mini
To analyze the relationship between the successive terms in the sequence –3.2, 4.8, –7.2, 10.8, we will look for a common difference and then check for a common ratio.

**Finding the Common Difference:**
The common difference is calculated by subtracting one term from the next one.

1. 4.8 - (-3.2) = 4.8 + 3.2 = 8.0
2. -7.2 - 4.8 = -7.2 - 4.8 = -12.0
3. 10.8 - (-7.2) = 10.8 + 7.2 = 18.0

Since the differences are not constant (8.0, -12.0, and 18.0), the terms do not have a common difference.

**Finding the Common Ratio:**
The common ratio is calculated by dividing one term by the previous term.

1. 4.8 / (-3.2) = -1.5
2. -7.2 / 4.8 = -1.5
3. 10.8 / (-7.2) = -1.5

Since the ratios are the same, the terms have a common ratio of -1.5.

Therefore, the correct description of the relationship between the successive terms is:

**The terms have a common ratio of –1.5.**
Answered by GPT-4o mini
To determine which sequences are arithmetic, we need to check if they have a common difference. An arithmetic sequence has the same difference between consecutive terms.

1. **Sequence: –5, 5, –5, 5, –5, …**
- Differences:
- 5 - (-5) = 10
- -5 - 5 = -10
- 5 - (-5) = 10
- No common difference. **Not arithmetic.**

2. **Sequence: 96, 48, 24, 12, 6**
- Differences:
- 48 - 96 = -48
- 24 - 48 = -24
- 12 - 24 = -12
- 6 - 12 = -6
- No common difference. **Not arithmetic.**

3. **Sequence: 18, 5.5, –7, –19.5, –32**
- Differences:
- 5.5 - 18 = -12.5
- -7 - 5.5 = -12.5
- -19.5 - (-7) = -12.5
- -32 - (-19.5) = -12.5
- Common difference is -12.5. **Arithmetic!**

4. **Sequence: –1, –3, –9, –27, –81**
- Differences:
- -3 - (-1) = -2
- -9 - (-3) = -6
- -27 - (-9) = -18
- -81 - (-27) = -54
- No common difference. **Not arithmetic.**

5. **Sequence: 16, 32, 48, 64, 80**
- Differences:
- 32 - 16 = 16
- 48 - 32 = 16
- 64 - 48 = 16
- 80 - 64 = 16
- Common difference is 16. **Arithmetic!**

### Summary:
The sequences that are **arithmetic** are:

- **18, 5.5, –7, –19.5, –32**
- **16, 32, 48, 64, 80**
Answered by GPT-4o mini
To determine which sequences are geometric, we need to check if they have a common ratio between consecutive terms. A geometric sequence has the same ratio between consecutive terms.

1. **Sequence: –2, –4, –6, –8, –10, …**
- Ratios:
- -4 / -2 = 2
- -6 / -4 = 1.5
- -8 / -6 = 1.333...
- -10 / -8 = 1.25
- No common ratio. **Not geometric.**

2. **Sequence: 16, –8, 4, –2, 1**
- Ratios:
- -8 / 16 = -0.5
- 4 / -8 = -0.5
- -2 / 4 = -0.5
- 1 / -2 = -0.5
- Common ratio is -0.5. **Geometric!**

3. **Sequence: –15, –18, –21.6, –25.92, –31.104, …**
- Ratios:
- -18 / -15 = 1.2
- -21.6 / -18 = 1.2
- -25.92 / -21.6 = 1.2
- -31.104 / -25.92 = 1.2
- Common ratio is 1.2. **Geometric!**

4. **Sequence: 4, 10.5, 17, 23.5, 30, …**
- Ratios:
- 10.5 / 4 = 2.625
- 17 / 10.5 ≈ 1.619
- 23.5 / 17 ≈ 1.382
- 30 / 23.5 ≈ 1.278
- No common ratio. **Not geometric.**

5. **Sequence: 625, 125, 25, 5, 1, …**
- Ratios:
- 125 / 625 = 0.2
- 25 / 125 = 0.2
- 5 / 25 = 0.2
- 1 / 5 = 0.2
- Common ratio is 0.2. **Geometric!**

### Summary:
The sequences that are **geometric** are:

- **16, –8, 4, –2, 1**
- **–15, –18, –21.6, –25.92, –31.104, …**
- **625, 125, 25, 5, 1, …**
Answered by GPT-4o mini
To find \( f(5) \) using the recursive formula \( f(n + 1) = f(n) - 2 \) and the initial condition \( f(1) = 18 \), we will calculate the values step by step.

1. Calculate \( f(2) \):
\[
f(2) = f(1) - 2 = 18 - 2 = 16
\]

2. Calculate \( f(3) \):
\[
f(3) = f(2) - 2 = 16 - 2 = 14
\]

3. Calculate \( f(4) \):
\[
f(4) = f(3) - 2 = 14 - 2 = 12
\]

4. Calculate \( f(5) \):
\[
f(5) = f(4) - 2 = 12 - 2 = 10
\]

Thus, the value of \( f(5) \) is \( \boxed{10} \).
Answered by GPT-4o mini
To determine which sequence could be partially defined by the recursive formula \( f(n + 1) = f(n) + 2.5 \) for \( n \geq 1 \), we need to find a sequence where the difference between consecutive terms is consistently \( 2.5 \).

Let's examine each option:

1. **Sequence: 2.5, 6.25, 15.625, 39.0625, …**
- Differences:
- \( 6.25 - 2.5 = 3.75 \)
- \( 15.625 - 6.25 = 9.375 \)
- \( 39.0625 - 15.625 = 23.4375 \)
- The differences are not constant. **Not defined by the formula.**

2. **Sequence: 2.5, 5, 10, 20**
- Differences:
- \( 5 - 2.5 = 2.5 \)
- \( 10 - 5 = 5 \)
- \( 20 - 10 = 10 \)
- The differences are not consistent. **Not defined by the formula.**

3. **Sequence: –10, –7.5, –5, –2.5, …**
- Differences:
- \( -7.5 - (-10) = 2.5 \)
- \( -5 - (-7.5) = 2.5 \)
- \( -2.5 - (-5) = 2.5 \)
- The differences are all \( 2.5 \). **Defined by the formula.**

4. **Sequence: –10, –25, 62.5, 156.25**
- Differences:
- \( -25 - (-10) = -15 \)
- \( 62.5 - (-25) = 87.5 \)
- \( 156.25 - 62.5 = 93.75 \)
- The differences are not consistent. **Not defined by the formula.**

### Conclusion:
The sequence that could be partially defined by the recursive formula \( f(n + 1) = f(n) + 2.5 \) is:

- **–10, –7.5, –5, –2.5, …**
Answered by GPT-4o mini
To determine the correct recursive formula for the given sequence \(3, -6, 12, -24, 48, \ldots\), we need to observe the relationship between the terms.

1. **First, calculate the ratio of successive terms:**
- \( f(2) = -6\)
- \( f(3) = 12\)
- \( f(4) = -24\)
- \( f(5) = 48\)

Calculating the ratios:
- From \(f(1)\) to \(f(2)\):
\[
f(2) = -6 = -2 \times 3
\]

- From \(f(2)\) to \(f(3)\):
\[
f(3) = 12 = -2 \times (-6)
\]

- From \(f(3)\) to \(f(4)\):
\[
f(4) = -24 = -2 \times 12
\]

- From \(f(4)\) to \(f(5)\):
\[
f(5) = 48 = -2 \times (-24)
\]

2. **Observe the pattern:**
- Each term is obtained from the previous term by multiplying by \(-2\). Therefore, the recursive relationship can be expressed as:

\[
f(n + 1) = -2f(n)
\]

### Conclusion:
The recursive formula that can be used to generate the sequence \(3, -6, 12, -24, 48, \ldots\) is:

\[
f(n + 1) = -2f(n).
\]

Thus, the correct answer is:

- **f(n + 1) = –2 f(n)**.