Sequence Practice Problems - Arithmetic & Geometric Patterns

Understanding Series / Sequences

Complete explanation with examples

What is a Sequence?

Mathematical sequences are a group of terms with a certain rule that dictates a certain operation must be performed and repeated in order to get from one term to the next.
The operation can be addition, subtraction, multiplication, division, or any other mathematical operation.

For example, the following is a basic numerical series:
$1, 2, 3, 4, 5$

To get from one term to the next in the sequence we add $+1$ .
$2 = 1+1$
$3 = 2+1$
$4 = 3+1$
And so on.

Using proper notation:

a₁ = 1 (first term)
a₂ = 2 (second term)
a₃ = 3 (third term)
a₄ = 4 (fourth term)
a₅ = 5 (fifth term)

The pattern: Add 1 to get the next term

a₂ = a₁ + 1 = 1 + 1 = 2
a₃ = a₂ + 1 = 2 + 1 = 3
a₄ = a₃ + 1 = 3 + 1 = 4
And so on...

This is called an arithmetic sequence because we add the same value (called the common difference) to get from one term to the next.

Key Characteristics of Sequences

Order Matters: Terms have specific positions

1st term, 2nd term, 3rd term, etc.
The position of each term is crucial to the sequence

Standard Notation:

We typically denote sequences using subscript notation: a₁, a₂, a₃, a₄, ...
aₙ represents the nth term (the general term)
The subscript indicates the position of the term

Pattern or Rule:

There's a relationship that connects the terms
This rule can involve addition, subtraction, multiplication, division, or more complex mathematical operations

Comparison of sequences: The arithmetic sequence starts at -6 and increases by 7 each time (-6, 1, 8, 15, 22). The geometric sequence starts at 1 and multiplies by 3 each time (1, 3, 9, 27, 81). Arrows indicate the operation between term

Detailed explanation

Practice Series / Sequences

Test your knowledge with 41 quizzes

Look at the following set of numbers and determine if there is any property, if so, what is it?

$$ 256,64,16,4,1 $$

Examples with solutions for Series / Sequences

Step-by-step solutions included

Exercise #1

12 ☐ 10 ☐ 8 7 6 5 4 3 2 1

Which numbers are missing from the sequence so that the sequence has a term-to-term rule?

Step-by-Step Solution

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: $8+1=9$

$10+1=11$

Answer:

11 , 9

Video Solution

Exercise #2

Is there a term-to-term rule for the sequence below?

18 , 22 , 26 , 30

Step-by-Step Solution

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.

Answer:

Yes

Video Solution

Exercise #3

Look at the following set of numbers and determine if there is any property, if so, what is it?

$1,2,3,4,5,6$

Step-by-Step Solution

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 $+1$ .

Answer:

$+1$

Video Solution

Exercise #4

Look at the following set of numbers and determine if there is any property, if so, what is it?

$10,8,6,4,2$

Step-by-Step Solution

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: $-2$ .

Answer:

$-2$

Video Solution

Exercise #5

Look at the following set of numbers and determine if there is any property, if so, what is it?

$94,96,98,100,102,104$

Step-by-Step Solution

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......

Answer:

$+2$

Video Solution

Frequently Asked Questions

Everything you need to know about Series / Sequences

How do you find the pattern in a sequence?

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To find a sequence pattern, examine the difference or ratio between consecutive terms. For arithmetic sequences, look for a constant difference (like +3 or -2). For geometric sequences, look for a constant ratio (like ×2 or ÷4). Practice with examples like 2, 4, 8, 16 (multiply by 2) or 6, 4, 2, 0 (subtract 2).

What is the difference between arithmetic and geometric sequences?

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Arithmetic sequences use addition or subtraction with a constant difference between terms (example: 1, 3, 5, 7 adds +2). Geometric sequences use multiplication or division with a constant ratio (example: 3, 9, 27, 81 multiplies by 3). Understanding this difference is key to solving sequence problems correctly.

How do you solve sequence word problems step by step?

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Follow these steps: 1) Write out the given terms in order, 2) Find the pattern by checking differences or ratios, 3) Verify the pattern works for all given terms, 4) Apply the rule to find missing terms, 5) Double-check your answer. Practice makes perfect with sequence problem solving.

What are the most common sequence patterns in math?

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The most common patterns include: • Arithmetic: constant addition/subtraction (2, 5, 8, 11) • Geometric: constant multiplication/division (4, 12, 36, 108) • Fibonacci: sum of previous two terms (1, 1, 2, 3, 5, 8) • Square numbers: 1, 4, 9, 16, 25 • Powers: 2, 4, 8, 16, 32

How do you find missing terms in a sequence?

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First identify the sequence rule by examining given terms. Then apply the rule systematically to fill gaps. For example, in the sequence 5, __, 17, 23, find the difference (6) and calculate the missing term: 5 + 6 = 11. Always verify your answer fits the overall pattern.

What is the Fibonacci sequence and how does it work?

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The Fibonacci sequence starts with 1, 1, then each following number is the sum of the two previous numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34... The rule is: F(n) = F(n-1) + F(n-2). This famous sequence appears in nature and has many mathematical applications.

Why are sequence problems important in mathematics?

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Sequence problems develop pattern recognition skills essential for algebra, calculus, and real-world problem solving. They teach logical thinking, help students understand mathematical relationships, and prepare them for advanced topics like functions and series. Mastering sequences builds a strong foundation for higher mathematics.

What are some tips for solving difficult sequence problems?

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Try these strategies: 1) Look for multiple operation patterns (add then multiply), 2) Check if terms follow formulas like n², 2ⁿ, or factorials, 3) Consider alternating patterns or signs, 4) Work backwards from known terms, 5) Draw diagrams when helpful, 6) Practice regularly with various sequence types to build recognition speed.

More Series / Sequences Questions

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Series / Sequences

Complete the Arithmetic Sequence: From 20 to 2 in Chair Numbers Find the Missing Term in Powers Sequence: 729, _, 81, 27, 9, 3 Arithmetic Sequence: Finding December's Food Price in 30→33→36 Pattern Arithmetic Sequence Problem: Finding Terms Before 10 with -3.5 Difference Find the 7th and 8th Terms in Sequence: 6, 16, 8, 18, 10, 20

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