Sequences

A sequence is just a list of numbers in an order defined by a specific pattern. 

There are three common types you’ll see most often:

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1. Arithmetic sequences → add or subtract the same number each time.
   
   
2. Geometric sequences → multiply (or divide) by the same number each time.
   
   
3. Other patterns → sometimes it’s “add 1, then 2, then 3…” or something less regular.
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Arithmetic Sequences

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Since an arithmetic sequence means adding or subtracting the same number each term, there is a general formula where one could find any term in the sequence. This can be referred to as the nth term. For example, if asked to find the 928th term, n is 928. 

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x_n = a + d(n−1)

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Where x_n is the value of the term, a is the first term, d is the difference or number it goes up or down by, and n is the number of the term.

You can also solve sequences by following the pattern, but having a general formula allows you to find any term of that sequence. 

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This is a video of a live teacher showing examples of arithmetic sequences:

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Example 1

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Question: What number comes next?
100, 94, 88, 82, 76, __

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Step 1: Look at the changes
100 → 94 (down 6)
94 → 88 (down 6)
88 → 82 (down 6)
82 → 76 (down 6)

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Step 2: Identify the rule
Subtract 6 each time. That makes this an arithmetic sequence.

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Step 3: Apply the rule
76 – 6 = 70

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Answer: 70

Teacher tip: Always check at least 2–3 jumps to be sure the pattern is consistent.

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Example 2

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Question: What is the 17th number of this sequence?
100, 94, 88, 82, 76, __

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Step 1: Look at the changes
100 → 94 (down 6)
94 → 88 (down 6)
88 → 82 (down 6)
82 → 76 (down 6)

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Step 2: Identify the rule
Subtract 6 each time. That makes this an arithmetic sequence.

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Step 3: Substitute the values into the general formula for the 17th term

x_n = a + d(n−1)

x₁₇ = 100 + (-6)(17−1)

= 100 + (-6)(16)

= 4

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Answer: 4

This is a Khan Academy video that walks through arithmetic sequences:

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Geometric Sequences

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Geometric sequences also have a general formula to find the nth term. The pattern is that the numbers in sequence are being multiplied by the same number. 

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x_n = ar^(n-1)

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Where a is the first term and r is the number being multiplied, or the “common ratio”

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Example 1

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Question: What number comes next?
3, 9, 27, 81, __

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Step 1: Look at the changes
3 → 9 (×3)
9 → 27 (×3)
27 → 81 (×3)

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Step 2: Identify the rule
Multiply by 3 each time. This is a geometric sequence.

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Step 3: Apply the rule
81 × 3 = 243

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Answer: 243

Teacher tip: If the numbers are growing really fast (or shrinking quickly), think multiplication/division rather than addition/subtraction.

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Example 2

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Question: What is the 20th number in this sequence?
3, 9, 27, 81, __

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Step 1: Look at the changes
3 → 9 (×3)
9 → 27 (×3)
27 → 81 (×3)

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Step 2: Identify the rule
Multiply by 3 each time. This is a geometric sequence.

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Step 3: Substitute the values into the general formula for the 20th term

x_n = ar^(n-1)

Where a is 3 and r is 3

x₂₀ = 3(3)^(20-1)

=3(3)¹⁹

= 3  1162261467

= 3486784401

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Answer: 3486784401

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Other Sequences

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Example 1

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Question: What number comes next?
1, 2, 4, 7, 11, 16, __

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Step 1: Look at the changes
1 → 2 (+1)
2 → 4 (+2)
4 → 7 (+3)
7 → 11 (+4)
11 → 16 (+5)

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Step 2: Identify the rule
We’re adding bigger and bigger numbers each time: +1, +2, +3, +4, +5…
This is a growing step sequence, not a simple arithmetic one.

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Step 3: Apply the rule
Next step is +6.
16 + 6 = 22

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Answer: 22

👉 Teacher tip: If the differences keep changing, check whether they’re increasing by 1 each time.

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Quick Summary

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• Arithmetic sequence: add or subtract the same number each time.
  Example: 5, 10, 15, 20 (rule = +5).
  
  
• Geometric sequence: multiply or divide by the same number each time.
  Example: 2, 4, 8, 16 (rule = ×2).
  
  
• Other patterns: watch the differences — sometimes they increase by 1 each time, or alternate.
  
  

Here is a video describing the differences between them:

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Practice Problems

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Try these:

1. 50, 45, 40, 35, __
2. 2, 6, 18, 54, __
3. 1, 3, 6, 10, 15, __
4. 128, 64, 32, 16, __
   
   

Answers:

1. 30 (subtract 5 each time)
2. 162 (×3 each time)
3. 21 (add +2, +3, +4, +5, +6 …)
4. (divide by 2 each time)

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Sequences: Harder Examples

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Example 1

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Question: What number comes next?
2, 5, 10, 17, 26, __

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Step 1: Look at the differences
5 – 2 = 3
10 – 5 = 5
17 – 10 = 7
26 – 17 = 9

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Step 2: Identify the rule
The differences are odd numbers: +3, +5, +7, +9…

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Step 3: Apply the rule
Next difference = +11.
26 + 11 = 37

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Answer: 37

This is a “quadratic sequence” — the differences themselves follow a pattern.

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Example 2

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Question: What number comes next?
1, 1, 2, 3, 5, 8, 13, ___

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Step 1: Look carefully
It looks like it is adding the previous number

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Step 2: Check for consistency
1 + 0, 1 +1, 2 + 1, 3 + 2 and so on

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Step 3: Follow the pattern

Add 8 and 13

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Answer: 21

This is also known as the Fibonacci sequence! 

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Example 3

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Question: What number comes next?
2, 6, 12, 20, 30, __

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Step 1: Look at the differences
6 – 2 = 4
12 – 6 = 6
20 – 12 = 8
30 – 20 = 10

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Step 2: Identify the rule
Differences are increasing by 2 each time (+4, +6, +8, +10…).

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Step 3: Apply the rule
Next difference = +12.
30 + 12 = 42

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Answer: 42

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Example 4

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Question: What number comes next?
81, 27, 9, 3, __

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Step 1: Look at the changes
81 ÷ 3 = 27
27 ÷ 3 = 9
9 ÷ 3 = 3

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Step 2: Apply the rule
Keep dividing by 3.
3 ÷ 3 = 1

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Answer: 1

This is geometric with a fraction multiplier (× 1/3).

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Key Takeaways for Harder Sequences

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1. Always check differences first (if they’re steady → arithmetic; if they’re growing → quadratic-type).
   
2. If differences don’t help, try ratios (geometric patterns).
   
3. If neither works, look for “previous term rules” (like Fibonacci: add last two).
   
   

Harder Practice for You

Try these:

1. 4, 9, 16, 25, 36, __
   
2. 1, 2, 6, 24, 120, __
   
3. 3, 7, 15, 31, 63, __
   
4. 2, 3, 5, 9, 17, 33, __
   
5. 10, 20, 40, 80, 160, 320, __
   
   

Answers:

1. 49 (perfect squares: 2², 3², 4², …)
   
2. 720 (factorials: 1!, 2!, 3!, 4!, 5!, 6!)
   
3. 127 (pattern is ×2 + 1)
   
4. 65 (each term is previous ×2 – 1)
   
5. 640 (geometric, ×2 each time)