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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.

11 answers

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.

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

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
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, …

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, …
A sequence is defined by the recursive formula f (n + 1) = f(n) – 2. If f(1) = 18, what is f(5)?

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 \( \boxed{10} \).

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

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, …
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)

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

  1. 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).