Sequence Practice Problems - Arithmetic & Geometric Patterns

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:$8+1=9$

$10+1=11$

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 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: $+5$.

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'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.