Series / Sequences: Identifying and defining elements

It is possible to see that there is a difference of one number between each number.

That is, 1 is added to each number and it will be the next number:

$1+1=2$

$2+1=3$

$3+1=4$

Etcetera. Therefore, the next numbers missing in the sequence will be:$8+1=9$

$10+1=11$

To solve this problem, we'll check the differences between consecutive terms:

• The difference between $22$ and $18$ is $22 - 18 = 4$.
• The difference between $26$ and $22$ is $26 - 22 = 4$.
• The difference between $30$ and $26$ is $30 - 26 = 4$.

All differences between consecutive terms are $4$, indicating a constant increment. Thus, the sequence is arithmetic with a common difference of $4$.

The term-to-term rule is: to get the next term, add $4$ to the current term.

Therefore, yes, there is a term-to-term rule for this sequence, given by adding $4$ to the previous term.

To solve this problem, we need to determine if there's a consistent pattern or rule in the sequence $1, 2, 3, 4, 5, 6$.

Let's proceed step by step:

• Step 1: Analyze the given sequence
  The sequence is $1, 2, 3, 4, 5, 6$.
• Step 2: Check for a common difference
  Calculate the difference between each consecutive pair of numbers:

$2 - 1 = 1$
$3 - 2 = 1$
$4 - 3 = 1$
$5 - 4 = 1$
$6 - 5 = 1$

From the calculations above, we observe that the difference between each consecutive term is $+1$.

Conclusion: The sequence is an arithmetic sequence with a common difference of $+1$.

Therefore, the correct choice is $+1$.

To solve this problem, we need to analyze whether the set of numbers $10, 8, 6, 4, 2$ has a pattern or property.

• Step 1: Observe the difference between consecutive terms:
  $8 - 10 = -2$
  $6 - 8 = -2$
  $4 - 6 = -2$
  $2 - 4 = -2$
• Step 2: Analyze the result.
  We see that the difference between consecutive terms is consistently $-2$.

This indicates that the terms form an arithmetic sequence with a common difference of $-2$.

Hence, the property of this set of numbers is that it is an arithmetic sequence with a common difference of $-2$.

By comparing the possible answer choices, we confirm that the correct choice is number 1: $-2$.

One can observe that the difference between each number is 2.

That is, with each leap the next number increases by 2:

$94+2=96$

$96+2=98$

$98+2=100$

and so forth......

To solve this problem, we'll check for consistent differences between the numbers, as this can indicate a property such as an arithmetic sequence.

• Step 1: Calculate the difference between each pair of consecutive numbers.

Let's look at the differences:

$16 - 13 = 3$

$20 - 16 = 4$

$23 - 20 = 3$

• Step 2: Analyze the differences.

The differences between consecutive numbers are not consistent: $3, 4,$ and $3$.

This irregularity shows that there is no single property like a consistent common difference, which would indicate an arithmetic sequence.

Therefore, no particular property applies to this set as a whole based on the differences analyzed.

The correct choice is that a regular property does not exist among these numbers.

Therefore, the solution to the problem is: Does not exist.

To solve this problem, we will determine if the given sequence of numbers follows a particular pattern or property:

First, we list the sequence provided: $13, 10, 7, 4, 1$.

Since an arithmetic sequence is one of the simplest patterns, we will check for a common difference, which involves subtracting each term from the next:

• Calculate the difference between the first and second terms: $10 - 13 = -3$.
• Calculate the difference between the second and third terms: $7 - 10 = -3$.
• Calculate the difference between the third and fourth terms: $4 - 7 = -3$.
• Calculate the difference between the fourth and fifth terms: $1 - 4 = -3$.

We observe that the difference between each consecutive pair of numbers is consistently $-3$. This implies that the sequence has a common difference of $-3$, and therefore, it is an arithmetic sequence.

In conclusion, the identified property for the sequence is that it is an arithmetic sequence with a common difference of $-3$.

Therefore, the solution to the problem is $-3$.

To solve this problem of finding the rule for the sequence $5, 10, 15, 20, 25, 30$, we will follow these steps:

• Step 1: Analyze the difference between consecutive numbers in the sequence.
• Step 2: Identify a consistent pattern or rule.
• Step 3: Compare the pattern against the given multiple-choice answers.

Now, let's work through each step:

Step 1: Calculate the difference between consecutive terms:

$10 - 5 = 5$

$15 - 10 = 5$

$20 - 15 = 5$

$25 - 20 = 5$

$30 - 25 = 5$

Step 2: We observe that the difference between each pair of successive numbers is $5$, which is consistent throughout the sequence.

Step 3: Compare this pattern with the given choices. The choice corresponding to adding 5 consistently matches our observed pattern.

Therefore, the rule for this sequence is to add 5 to each preceding number to obtain the next number in the sequence. This corresponds with choice number 2: $+5$.

It is possible to see that if you multiply each number from the right column by 2, you get the number from the left column.

That is:$1\times2=2$

$2\times2=4$

$3\times2=6$

Therefore, the number of balls is 2 times greater than the number of courts.

To solve this problem, let's analyze the sequence of numbers given: $2, 4, 8, 16, 32, 64$.

First, let's calculate the ratio between each pair of consecutive terms in the sequence:

• The ratio of the second term $4$ to the first term $2$ is $\frac{4}{2} = 2$.
• The ratio of the third term $8$ to the second term $4$ is $\frac{8}{4} = 2$.
• The ratio of the fourth term $16$ to the third term $8$ is $\frac{16}{8} = 2$.
• The ratio of the fifth term $32$ to the fourth term $16$ is $\frac{32}{16} = 2$.
• The ratio of the sixth term $64$ to the fifth term $32$ is $\frac{64}{32} = 2$.

Each consecutive term is obtained by multiplying the previous term by $2$. Therefore, this sequence is a geometric sequence where each term is $\times 2$ times the preceding term.

This consistent multiplication factor shows that the sequence's property is that of a geometric progression with a common ratio $r = 2$.

In conclusion, the identified property of the sequence is that each term is multiplied by $2$, which is option $\times 2$.

To solve this problem, we'll follow these steps:

• Step 1: Check if the sequence is an arithmetic sequence.
• Step 2: Check if the sequence is a geometric sequence.
• Step 3: Explore other patterns if neither arithmetic nor geometric sequence fits.

Now, let's work through each step:

Step 1: Checking for arithmetic sequence:
To be an arithmetic sequence, the difference between consecutive terms should be constant.
For $1, 3, 9, 26, 81$:
$3 - 1 = 2$
$9 - 3 = 6$
$26 - 9 = 17$
$81 - 26 = 55$
The differences are not the same, so it is not an arithmetic sequence.

Step 2: Checking for geometric sequence:
To be a geometric sequence, the ratio of consecutive terms should be constant.
For $1, 3, 9, 26, 81$:
$\frac{3}{1} = 3$
$\frac{9}{3} = 3$
$\frac{26}{9}$ is not 3, similarly $\frac{81}{26}$ is not constant.
Thus, it is not a geometric sequence either.

Step 3: Explore other non-trivial patterns:
Without straightforward arithmetic or geometric patterns identified, try other patterns or sequences, but given the options provided, none of the numerical patterns visibly connect through such standard sequences. This suggests examining deeper provides diminishing returns without knowing additional context or rules that these numbers follow.

Evaluating choices: None of the options directly covers a standard pattern or rule fitting all these points consistently, indicating no obvious, identical property unifies the set.

Therefore, the correct answer is Does not exist, as there isn't an evident mathematical property or pattern connecting these numbers uniformly.

Therefore, the solution to the problem is Does not exist.

To identify the property of the sequence $256, 64, 16, 4, 1$, we'll calculate the ratio between each consecutive pair of terms:

• First, calculate the ratio of the second term to the first term: $\frac{64}{256} = \frac{1}{4}$.
• Next, calculate the ratio of the third term to the second term: $\frac{16}{64} = \frac{1}{4}$.
• Then, calculate the ratio of the fourth term to the third term: $\frac{4}{16} = \frac{1}{4}$.
• Finally, calculate the ratio of the fifth term to the fourth term: $\frac{1}{4} = \frac{1}{4}$.

All calculated ratios are equal to $\frac{1}{4}$. This confirms that each term in the sequence is obtained by multiplying the previous term by $\frac{1}{4}$ or equivalently, multiplying by 0.25.

Therefore, the sequence follows the property of being a geometric sequence with a common ratio of $0.25$.

This corresponds to the answer choice: $\times 0.25$ (Choice 4).

To solve this problem, we'll inspect the differences between consecutive numbers:

First, calculate the differences:

• $24 - 12 = 12$
• $35 - 24 = 11$
• $48 - 35 = 13$
• $60 - 48 = 12$

The differences are $12, 11, 13,$ and $12$.

We observe that there is no consistent pattern or common difference in these numbers, which implies the numbers do not form a regular arithmetic or geometric sequence.

Therefore, the property does not consistently exist across all numbers.

Consequently, the solution to this problem is that a common property does not exist.

To solve this problem, we will investigate whether the sequence of numbers $4, 8, 12, 5, 20$ possesses any identifiable pattern or property, focusing primarily on identifying an arithmetic pattern.

Let's examine the differences between consecutive terms:

• The difference between $8$ and $4$ is $8 - 4 = 4$.
• The difference between $12$ and $8$ is $12 - 8 = 4$.
• The difference between $5$ and $12$ is $5 - 12 = -7$.
• The difference between $20$ and $5$ is $20 - 5 = 15$.

We observe that the differences between consecutive numbers are not consistent. Since the sequence does not exhibit a constant difference, it is not an arithmetic sequence.

We could also consider checking for a geometric pattern, but since immediate calculations show variations in both differences and potential ratios, this seems unnecessary for a basic sequence like this.

None of the properties we considered (arithmetic or geometric) apply. Thus, we conclude there is no identifiable pattern or property consistent across the entire sequence of numbers $4, 8, 12, 5, 20$.

Therefore, the correct answer is: Does not exist.

Let's analyze the sequence: $768, 192, 48, 12, 3$.

First, observe the relationship between consecutive terms by dividing one term by the previous term:

• $\frac{192}{768} = \frac{1}{4}$
• $\frac{48}{192} = \frac{1}{4}$
• $\frac{12}{48} = \frac{1}{4}$
• $\frac{3}{12} = \frac{1}{4}$

Each division yields the same factor: $\frac{1}{4}$.

This indicates a consistent operation of dividing by $4$ between the terms. Thus, the rule in the sequence is to divide the current term by $4$ to obtain the next term.

Therefore, this sequence follows the rule of dividing each number by $4$ to get the next number.

Thus, the correct choice that corresponds to this pattern is: $\text{Yes, multiply by 4.}$

To determine which sequence has no consistent pattern or property, we will examine each of the given sequences for an arithmetic progression. An arithmetic progression is defined by a sequence in which each term after the first is obtained by adding a constant difference to the previous term.

• Option 1: $6, 5, 4, 3, 2, 1$ - The differences are $-1, -1, -1, -1, -1$. This follows an arithmetic progression with a common difference of $-1$.
• Option 2: $2, 1, 0, 1, 2, 3$ - The differences are $-1, -1, +1, +1, +1$. The differences are inconsistent across the entire sequence, suggesting this does not follow a simple arithmetic progression.
• Option 3: $55, 45, 35, 30, 20, 0$ - The differences are $-10, -10, -5, -10, -20$. The differences are inconsistent, indicating no uniform arithmetic progression property is present throughout.
• Option 4: $60, 50, 40, 30, 20, 10$ - The differences are $-10, -10, -10, -10, -10$. This sequence follows an arithmetic progression with a constant difference of $-10$.

From this analysis, Option 3 does not follow a consistent arithmetic progression or property, as the differences between terms are inconsistent.

Therefore, the sequence in which there is no property is $55, 45, 35, 30, 20, 0$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the type of sequence for each option.
• Step 2: Apply the pattern to see if it holds throughout the group.

Now, let's analyze each choice:

Option 1: $10, 4, 3, 2, 1, 3$
The differences between each consecutive term are: $4 - 10 = -6$, $3 - 4 = -1$, $2 - 3 = -1$, $1 - 2 = -1$, $3 - 1 = 2$.
These differences are not consistent, indicating no consistent pattern. Thus, this is not an arithmetic sequence.

Option 2: $62, 58, 54, 50, 46, 42$
The differences between each consecutive term are: $58 - 62 = -4$, $54 - 58 = -4$, $50 - 54 = -4$, $46 - 50 = -4$, $42 - 46 = -4$.
These differences are all consistent, indicating an arithmetic sequence with a common difference of $-4$.

Option 3: $13, 9, 10, 10, 10, 1$
The differences between each consecutive term are: $9 - 13 = -4$, $10 - 9 = 1$, $10 - 10 = 0$, $10 - 10 = 0$, $1 - 10 = -9$.
These differences are not consistent, indicating no consistent pattern.

Option 4: $230, 200, 130, 100, 30, 26$
The differences between each consecutive term are: $200 - 230 = -30$, $130 - 200 = -70$, $100 - 130 = -30$, $30 - 100 = -70$, $26 - 30 = -4$.
These differences are not consistent, indicating no consistent pattern.

Therefore, the correct group that maintains the veracity as an arithmetic sequence is with a common difference of -4:

62, 58, 54, 50, 46, 42

To solve this problem, we'll follow these steps:

• Step 1: Examine differences between consecutive terms
• Step 2: Check for consistency in differences

Let's calculate the differences:

$85 - 64 = 21$

$98 - 85 = 13$

$100 - 98 = 2$

$1 - 100 = -99$

Observing these differences, it is evident that they are not consistent. Therefore, there is no consistent arithmetic or geometric pattern that applies to the entire sequence. Given the choices, the conclusion is clear: There is no term-to-term rule.

Therefore, the correct answer choice is 4: No.

To solve this problem, we'll follow these steps:

• Step 1: Identify the starting price and the weekly increase.
• Step 2: Use the arithmetic sequence formula to determine the price for each week.
• Step 3: Match the calculated sequence to the provided options to find the correct one.

Now, let's work through these steps:

Step 1: The shirt starts at a price of $20. Each week, the price increases by$5.

Step 2: Using the arithmetic sequence formula: $a_n = a_1 + (n-1) \cdot d$. - Week 0 (initial purchase): $a_1 = 20$ - Week 1: $a_2 = 20 + 1 \cdot 5 = 25$ - Week 2: $a_3 = 20 + 2 \cdot 5 = 30$ - Week 3: $a_4 = 20 + 3 \cdot 5 = 35$

Step 3: The calculated sequence is 20, 25, 30, 35, which needs to be reversed to match how sequences are usually listed.

The sequence in decreasing order is: $35, 30, 25, 20$.

This sequence matches choice 4, which is: 35, 30, 25, 20.

Therefore, the appropriate sequence to represent the price of the shirt is $35, 30, 25, 20$.