Recursive Formula Calculator – Find Recursive Rule

Recursive Formula Calculator

To calculate terms in a Recursive Sequence, input Initial Term (a₁), Recursive Rule, and Number of Terms desired. The calculator supports common recursive patterns including Arithmetic, Geometric, and Fibonacci-like sequences.

Select the Sequence Type and input parameters to generate sequence terms and identify patterns.

How to Find Recursive Rule

To find a Recursive Rule, first identify pattern by examining consecutive terms in the sequence. Look for Common Differences (arithmetic), Common Ratios (geometric), or other relationships between terms. For Arithmetic Sequences, subtract consecutive terms to find a constant difference. For Geometric Sequences, divide consecutive terms to find a constant ratio. For Complex Patterns, examine how each term relates to one or more previous terms.

Once you identify te pattern, write rule using proper notation. An Arithmetic Rule has the form aₙ = aₙ₋₁ + d, where d is the common difference. A Geometric Rule has the form aₙ = aₙ₋₁ × r, where r is the common ratio. For Combined Patterns, write how the current term depends on previous terms, such as aₙ = aₙ₋₁ + aₙ₋₂.

Always include the Initial Terms and state when n starts in your recursive rule. Verify your rule by checking if it correctly generates the next several terms in the sequence. Test the rule with known values to ensure it works for all cases.

How to Find Recursive Formula?

To find a Recursive Formula, analyze relationship between consecutive terms. Look for constant differences (arithmetic), constant ratios (geometric), or patterns involving previous terms. The formula requires an Initial Value and a Rule showing how to find the next term using previous terms. Common patterns include adding/subtracting (arithmetic), multiplying/dividing (geometric), or combining previous terms.

Formula for Recursive Calculations

Arithmetic Sequence:

aₙ = aₙ₋₁ + d
where:
aₙ = Current term
aₙ₋₁ = Previous term
d = Common difference
a₁ = First term (initial value)

Geometric Sequence:

aₙ = aₙ₋₁ × r
where:
aₙ = Current term
aₙ₋₁ = Previous term
r = Common ratio
a₁ = First term (initial value)

Fibonacci-like Sequence:

aₙ = aₙ₋₁ + aₙ₋₂
where:
aₙ = Current term
aₙ₋₁ = Previous term
aₙ₋₂ = Term before previous
a₁, a₂ = Initial values

Example 1: Arithmetic Sequence

  • Initial Term = 3
  • Common Difference = 5
  • First 5 Terms = 3, 8, 13, 18, 23
  • Rule = aₙ = aₙ₋₁ + 5
  • Type = Arithmetic Growth

Example 2: Geometric Sequence

  • Initial Term = 2
  • Common Ratio = 3
  • First 5 Terms = 2, 6, 18, 54, 162
  • Rule = aₙ = aₙ₋₁ × 3
  • Type = Geometric Growth

Example 3: Fibonacci Sequence

  • Initial Terms = 1, 1
  • Rule = aₙ = aₙ₋₁ + aₙ₋₂
  • First 5 Terms = 1, 1, 2, 3, 5
  • Type = Sum of Previous Two
  • Pattern = Golden Ratio Growth

Example 4: Custom Sequence

  • Initial Term = 4
  • Rule = aₙ = 2aₙ₋₁ – 1
  • First 5 Terms = 4, 7, 13, 25, 49
  • Type = Linear Recursion
  • Pattern = Exponential Growth

Example 5: Alternating Sequence

  • Initial Terms = 2, -2
  • Rule = aₙ = -aₙ₋₁
  • First 5 Terms = 2, -2, 2, -2, 2
  • Type = Alternating Pattern
  • Pattern = Oscillating Values

What is Recursive Rule Formula?

A Recursive Rule Formula defines how to find next term in a sequence using one or more previous terms. It consists of an Initial Value (or values) and a Rule that shows the relationship between terms. The formula allows you to generate sequence terms recursively, meaning each new term depends on previously calculated terms. Common types include Arithmetic (constant addition/subtraction), Geometric (constant multiplication/division), and Mixed patterns combining multiple operations or previous terms.

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