Ask a New Question
Menu

Which of the following statements correctly determines the rule for generating the next term in the sequence 23, 19.5, 16, 12.5....(1 point)

Responses

Multiply by the common ration of -3.5
Multiply by the common ration of -3.5

Add the common difference of -3.5
Add the common difference of -3.5

Add the common ratio of -3.5
Add the common ratio of -3.5

Multiply by the common difference of -3.5

9 answers

The given sequence is 23, 19.5, 16, 12.5. To determine the rule for generating the next term, we can look at the differences between consecutive terms:

  • From 23 to 19.5, the difference is 19.5 - 23 = -3.5.
  • From 19.5 to 16, the difference is 16 - 19.5 = -3.5.
  • From 16 to 12.5, the difference is 12.5 - 16 = -3.5.

Since the difference between each consecutive term is consistently -3.5, the rule for generating the next term is to add the common difference of -3.5.

Therefore, the correct statement is:

Add the common difference of -3.5.

Classify the following list of numbers as an arithmetic sequence, a geometric sequence, or neither.

9, 4, -1, -6

(1 point)
Responses

Neither
Neither

Geometric
Geometric

Arithmetic
Arithmetic

Not enough information given

To classify the sequence 9, 4, -1, -6, we need to determine if it is an arithmetic sequence, a geometric sequence, or neither.

  1. Arithmetic Sequence: In an arithmetic sequence, the difference between consecutive terms is constant.

    • First difference: 4 - 9 = -5
    • Second difference: -1 - 4 = -5
    • Third difference: -6 - (-1) = -5

    Since the difference is the same (-5) for all consecutive terms, this is an arithmetic sequence.

  2. Geometric Sequence: In a geometric sequence, the ratio of consecutive terms is constant.

    • First ratio: 4 / 9 ≈ 0.444
    • Second ratio: -1 / 4 = -0.25
    • Third ratio: -6 / -1 = 6

    The ratios are not constant, so it is not a geometric sequence.

Since the sequence has a constant difference, it is classified as:

Arithmetic.

What is the common ratio in the following geometric sequence?

2, 1/2,1/8,1/32...


(1 point)
Responses

-1.5
-1.5

1/4
1 fourth

1/2
1 half

4

To find the common ratio in the geometric sequence 2, 1/2, 1/8, 1/32, we divide each term by the previous term:

  1. From 2 to 1/2: \[ \text{Common ratio} = \frac{1/2}{2} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]

  2. From 1/2 to 1/8: \[ \text{Common ratio} = \frac{1/8}{1/2} = \frac{1}{8} \cdot \frac{2}{1} = \frac{2}{8} = \frac{1}{4} \]

  3. From 1/8 to 1/32: \[ \text{Common ratio} = \frac{1/32}{1/8} = \frac{1}{32} \cdot \frac{8}{1} = \frac{8}{32} = \frac{1}{4} \]

Since the common ratio is constant and equal to \( \frac{1}{4} \) for each pair of terms, the correct response is:

1/4.

Given the first five terms of the sequence 19, 15, 11, 7, 3... use the subscript notation to describe a7
.(1 point)
Responses

a7=−5
A sub 7 is equal to negative 5

a7=−9
A sub 7 is equal to negative 9

a7=−1
A sub 7 is equal to negative 1

a7=−4

To find \( a_7 \) in the sequence given (19, 15, 11, 7, 3...), we first need to determine the pattern of the sequence.

Step 1: Identify the common difference

The differences between consecutive terms are as follows:

  • \( 15 - 19 = -4 \)
  • \( 11 - 15 = -4 \)
  • \( 7 - 11 = -4 \)
  • \( 3 - 7 = -4 \)

The common difference is \( -4 \).

Step 2: Using the formula for the \( n \)-th term of an arithmetic sequence

The formula for the \( n \)-th term of an arithmetic sequence can be written as: \[ a_n = a_1 + (n - 1) \cdot d \] Where:

  • \( a_1 \) = first term
  • \( d \) = common difference
  • \( n \) = term number

In this case:

  • \( a_1 = 19 \)
  • \( d = -4 \)
  • We want to find \( a_7 \).

Step 3: Calculate \( a_7 \)

\[ a_7 = 19 + (7 - 1) \cdot (-4) \] \[ a_7 = 19 + 6 \cdot (-4) \] \[ a_7 = 19 - 24 \] \[ a_7 = -5 \]

Thus, \( a_7 = -5 \).

The correct response is: a7 = −5 (A sub 7 is equal to negative 5).

Given the first five terms of the sequence 81, 27, 9, 3, 1, 13
... use the subscript notation to describe a7
.(1 point)
Responses

a7=27
A sub 7 is equal to 27

a7=1/9
A sub 7 is equal to 1 ninth

a7=3
A sub 7 is equal to 3

a7=1/27

This sequence is \( 81, 27, 9, 3, 1, \ldots \). To find the value of \( a_7 \), we first need to determine whether the sequence is arithmetic or geometric.

Step 1: Identify the type of sequence

  1. Check if it is geometric: Calculate the ratio between consecutive terms:

    • \( \frac{27}{81} = \frac{1}{3} \)
    • \( \frac{9}{27} = \frac{1}{3} \)
    • \( \frac{3}{9} = \frac{1}{3} \)
    • \( \frac{1}{3} = \frac{1}{3} \)

    The common ratio is consistent: \( r = \frac{1}{3} \).

Step 2: Use the formula for the \( n \)-th term of a geometric sequence

The formula for the \( n \)-th term of a geometric sequence is given by: \[ a_n = a_1 \cdot r^{(n-1)} \]

Where:

  • \( a_1 \) is the first term
  • \( r \) is the common ratio
  • \( n \) is the term number

Step 3: Calculate \( a_7 \)

In this case:

  • \( a_1 = 81 \)
  • \( r = \frac{1}{3} \)
  • We want to find \( a_7 \).

Using the formula: \[ a_7 = 81 \cdot \left(\frac{1}{3}\right)^{7-1} \] \[ a_7 = 81 \cdot \left(\frac{1}{3}\right)^{6} \] \[ = 81 \cdot \frac{1}{729} \] \[ = \frac{81}{729} \] \[ = \frac{1}{9} \]

Therefore, \( a_7 = \frac{1}{9} \).

The correct response is: a7 = 1/9 (A sub 7 is equal to 1 ninth).