Sequences / Skips up to 100 - Examples, Exercises and Solutions

To solve this problem, we need to identify the pattern in the sequence provided, which initially lists the numbers $1$ and $3$.

First, observe the given numbers: $1$ and $3$.

The difference between the first term $1$ and the second term $3$ is:
$3 - 1 = 2$.

This suggests a common difference of $2$, implying that the sequence is likely an arithmetic sequence where each term increases by $2$.

We can use this observation to predict the next terms in the sequence:

• Starting with $1$, add the common difference $2$:
  $1 + 2 = 3$.
• From $3$, add the common difference $2$:
  $3 + 2 = 5$.
• Continuing this pattern, from $5$, add $2$:
  $5 + 2 = 7$.
• And from $7$, add the common difference $2$:
  $7 + 2 = 9$.

Thus, the sequence can be extended as:
$1, 3, 5, 7, 9$.

From the possible choices, the correct sequence is represented by choice 4.

Therefore, the correct answer to the problem is $1, 3, 5, 7, 9$.

To solve this problem, we'll identify whether the sequence follows a recognizable pattern. The sequence so far is $5, 7$.

Let's determine whether it follows an arithmetic sequence:

• First Term ($a_1$) = 5
• Second Term ($a_2$) = 7
• The Common Difference ($d$) = $7 - 5 = 2$

Assuming a consistent common difference, the sequence appears to be an arithmetic sequence increasing every term by 2.

Let's calculate the next few terms:

• Third Term ($a_3$) = $7 + 2 = 9$
• Fourth Term ($a_4$) = $9 + 2 = 11$
• Fifth Term ($a_5$) = $11 + 2 = 13$

Thus, the full sequence is: $5, 7, 9, 11, 13$.

A review of the multiple-choice options shows that the correct answer is given by choice 1: $5, 7, 9, 11, 13$.

Therefore, the complete sequence is $5, 7, 9, 11, 13$.

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

• Step 1: Determine the common difference by looking at the known terms, 21 and 19.
• Step 2: Apply the common difference to find the missing terms in the sequence.
• Step 3: Verify the sequence pattern to ensure accuracy.

Now, let's work through each step:

Step 1: The common difference between 21 and 19 is 2. Thus, each number in the sequence is reduced by 2 from the previous one.

Step 2: Starting from 25, subtract 2 to fill in the first gap: $25 - 2 = 23$. Now the sequence is 25, 23, ..., 21, 19.

From 19, subtract 2 to fill in the next gap: $19 - 2 = 17$, then $17 - 2 = 15$. Thus, the complete sequence is:

$25, 23, 21, 19, 17, 15$

Therefore, the solution to the problem is $25, 23, 21, 19, 17, 15$.

To solve the problem of completing the sequence $20, \ldots, 24, 26, \ldots, \ldots$, we recognize the sequence's underlying pattern.

Step 1: Analyze known terms.
The given sequence begins with $20, \ldots, 24, 26, \ldots, \ldots$. Notice the known terms $20$ and $24$, and $26$.

Step 2: Determine the difference between known terms.
The difference between $24$ and $20$ is $4$, suggesting an ongoing pattern.
Between $26$ and $24$, the difference is $2$, proposing alternation in differences or a complete oversight of interspersed terms around $26$.

Step 3: Determine full sequence consistency.
Check if numbers align with a two-step even-number sequence, which implies adding $2$ successively.
Extending forward and back confirms $20, 22, 24, 26, 28, \ldots$.

Step 4: Verification considering other instructions.
The sequence appears to be a straightforward arithmetic one comprising purely even numbers beginning from $20$. This step requires confirming subsequent numbers $28, 30$.

Conclusion: Sequential confirmation proves the arithmetic nature of the understanding, yielding:

$20,22,24,26,28,30$

To complete the given sequence $36, 34, \ldots$, we need to identify the pattern in the sequence. From the given terms, it appears that the sequence is decreasing.

Let's check if this is an arithmetic sequence, where each term decreases by a constant amount:

• Subtract the second term from the first term: $36 - 34 = 2$.
• This indicates that each term is decreasing by 2.

Recognizing this pattern, the sequence can be continued by subtracting 2 from each subsequent term:

• The next term after 34 is calculated as follows: $34 - 2 = 32$.
• Continuing, the term after 32 is: $32 - 2 = 30$.
• Finally, the term following 30 is: $30 - 2 = 28$.

Therefore, the complete sequence is $36, 34, 32, 30, 28$.

The correct answer choice, which matches this sequence, is:

$36,34,32,30,28$

The given sequence is $1113, 1112, 1111$.

Let's analyze the sequence:

• The first term is $1113$.
• The second term is $1112$, which is $1113 - 1$.
• The third term is $1111$, which is $1112 - 1$.

It's evident that each term is decreasing by $1$ from the previous term. Therefore, this sequence is an arithmetic sequence with a common difference of $-1$.

Given this information, we can continue the sequence by subtracting 1 from the last given term, $1111$.

• The next term is $1111 - 1 = 1110$.
• Following that, $1110 - 1 = 1109$.
• Finally, $1109 - 1 = 1108$.

Thus, the next three terms in the sequence are $1110, 1109,$ and $1108$.

Looking at the provided options, choice 4: $1110, 1109, 1108$, is the correct continuation of the sequence.

To solve this problem, we'll identify the pattern in the sequence:

• Step 1: Identify the order and pattern of the sequence
• Step 2: Follow the arithmetic pattern of adding the same number to the last term to find the next numbers
• Step 3: List the next terms in the sequence

Now, let's work through each step:

Step 1: The sequence given is 1007, 1008, 1009. We observe that each number is 1 more than the previous one.
Step 2: Continuing this pattern, the next number after 1009 is $1009 + 1 = 1010$.
Step 3: Similarly, we calculate the next numbers: $1010 + 1 = 1011$ and $1011 + 1 = 1012$.

Therefore, the sequence can be completed as: $1010, 1011, 1012$.

To solve this problem, we will continue the sequence $20,000, 20,001, 20,002, \ldots$ by adding 1 to the last number provided.

• Start with the last number in the sequence, which is $20,002$.
• Step 1: Add 1 to $20,002$ to get $20,003$.
• Step 2: Add 1 to $20,003$ to get $20,004$.
• Step 3: Add 1 to $20,004$ to get $20,005$.

Thus, the continuation of the sequence is $20,003, 20,004, 20,005$.

Therefore, the correct answer is choice 1: $20,003,20,004,20,005$.

To solve this problem, we will identify and use the pattern within the given sequence:

• Step 1: Start with the given numbers $60$ and $50$.
• Step 2: Calculate the difference between these two numbers. We have $60 - 50 = 10$.
• Step 3: Assume this difference will continue throughout the sequence. This assumption is necessary because it suggests a common arithmetic sequence structure.

Following these steps, we can complete the sequence:

First term is $60$.

Second term is $50$.

Based on the assumption of an arithmetic sequence with a common difference of $-10$:

Third term: $50 - 10 = 40$.

Fourth term: $40 - 10 = 30$.

Continuing this pattern:
Fifth term: $30 - 10 = 20$.
Sixth term: $20 - 10 = 10$.

Therefore, the complete sequence is $60, 50, 40, 30, 20, 10$.

To complete the sequence $20, 30, 40, \ldots$, we need to determine the pattern.

• Identify the pattern: Find the difference between successive terms.
• Calculate the difference: $30 - 20 = 10$ and $40 - 30 = 10$. The common difference is 10.
• Continue the sequence using this pattern: Add 10 to the last given number to find the next number.
• Extend the sequence:
• The term after 40 is $40 + 10 = 50$.
• The next term is $50 + 10 = 60$.
• The next term is $60 + 10 = 70$.

Thus, the complete sequence is $20, 30, 40, 50, 60, 70$.

Therefore, the solution to the problem is $20, 30, 40, 50, 60, 70$, corresponding to choice 4.

To solve this problem, let's follow these steps:

• Step 1: Calculate the differences between consecutive terms of the sequence.
• Step 2: Use the common difference to find the next terms.
• Step 3: Verify the result with the provided answer choices.

Step 1: Calculate the differences:
The difference between the first and second terms is $1000 - 955 = 45$.
The difference between the second and third terms is $1005 - 1000 = 5$.

Step 2: Identifying that the sequence alternates increases, first by 45 and then by 5, we predict it continues by adding 5. Thus, extending the sequence, the next terms would be:
$1005 + 5 = 1010$,
$1010 + 5 = 1015$,
$1015 + 5 = 1020$.

Therefore, the next terms in the sequence are $1010, 1015, 1020$.

Step 3: Verify against provided answer choices. The correct choice is:

$1010,1015,1020$

The next numbers in the sequence are $1010, 1015, 1020$.