Find the Recursive Formula Game/Worksheet

Find the Recursive Formula Game/Worksheet
In this game the player is presented with a generated sequence of numbers and must select the correct recursive formula that defines it. Scroll down the page for more details.

 

 

How to Play The Game

1. Observe the Stream
   When a sequence node boots up, look inside the main terminal card. You will see the first five numbers of an infinite pattern (e.g., 5, 12, 19, 26, 33…) alongside its baseline anchor index Given: a₁ = 5.
   

2. Isolate the Iteration Engine
   Analyze how each number transforms into the next. Ask yourself:
   Is it rising by a fixed amount? (Arithmetic spacing)
   Is it expanding exponentially? (Geometric scale)
   Does it rely on adding its immediate neighbors together? (Fibonacci architecture)
   

3. Evaluate the Choices
   Four multiple-choice mathematical formulas are listed in the dashboard grid. Every expression is structured in standard recursive notation where a_n-1 signifies the value of the previous term. Mentally run the numbers through each option.
   

4. Decode the Node
   Click on your chosen formula to verify.
   Correct Action: The card flashes glowing green, rewarding you with 300 points and moving your streak index upward.
   Incorrect Action: The card flashes red while highlighting the correct relation in green. An algorithmic breakdown unrolls below to demonstrate exactly how the formula builds the matching list.
   Click Next Sequence Challenge to clear the terminal cache and parse a new array.
   

The Underlying Mathematics
To successfully reverse-engineer a sequence into a recursive formula, you must identify its functional change model. The math embedded within this game’s database tracks four structural types of step transformations.

1. Constant Finite Differences (Arithmetic Type)
   If the step gaps between numbers remain uniformly flat, the recursive rule adds a constant value (d) to the previous value.
   

\(\Delta = a_n - a_{n-1} = d \implies a_n = a_{n-1} + d\)

Game Example: 5, 12, 19, 26, 33…
Analysis: 12 - 5 = 7; 19 - 12 = 7; 26 - 19 = 7.
Recursive Formula: a_n = a_n-1 + 72.

2. Constant Scaling Factors (Geometric Type)
   If the difference between consecutive items increases rapidly but their quotients remain fixed, the sequence scales by a geometric ratio (r).
   

\(\frac{a_n}{a_{n-1}} = r \implies a_n = r \cdot a_{n-1}\)

Game Example: 3, 6, 12, 24, 48…
Analysis: \(\frac{6}{3} = 2\); \(\frac{12}{6} = 2\); \(\frac{24}{12} = 2\).
Recursive Formula: a_n = 2a_n-1

3. Composite Operations (Linear Combinations)
   Some formulas multiply the previous term by a scale factor and apply an offset step at the same time.
   

a_n = m · a_n-1 + b

Game Example: 1, 4, 13, 40, 121…
Analysis: The differences (3, 9, 27, 81) are powers of 3, implying a tripling effect. Checking terms: 3(1) + 1 = 4; 3(4) + 1 = 13; 3(13) + 1 = 40.
Recursive Formula: a_n = 3a_n-1 + 1

4. Index-Dependent Functions (Non-Constant Changes)
   When changes alter based on where you are inside the sequence, the modification rule loops in the active index counter variable (n).
   

a_n = a_n-1 + f(n)

Game Example: 5, 7, 10, 14, 19…
Analysis: Let’s calculate the gaps between adjacent positions:
To get a₂ (7) from a₁ (5), we add 2. Notice that at this step, n = 2.
To get a₃ (10) from a₂ (7), we add 3. Notice that at this step, n = 3.
To get a₄ (14) from a₃ (10), we add 4. Notice that at this step, n = 4.

Recursive Formula: a_n = a_n-1 + n

Recursive Sequence Formulas

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