Recursive Rule Formula Calculator

Our Recursive Rule Formula Calculator is a powerful mathematical calculator that helps you generate and analyze sequences where each term depends on previous terms.

Our calculator handles three types of recursive sequences:

• Arithmetic sequences where you add the same value each time
• Geometric sequences where you multiply by the same value each time
• Custom recursive rules where new terms depend on two previous terms

Imagine you’re tracking monthly savings with a plan to increase your deposits by $50 each month. Starting with $100 in January, you’d have $150 in February, $200 in March, and so on. This is an arithmetic sequence with first term a₁ = 100 and common difference d = 50. The recursive rule is a_n = a_n-1 + 50.

Recursive Rule Formula?

A recursive rule formula defines how to find the next term in a sequence based on previous terms.

The three main types of recursive formulas are:

• Arithmetic Sequence: Each term differs from the previous by a constant value
  • Formula: a_n = a_n-1 + d
  • Where a_n is the current term, a_n-1 is the previous term, and d is the common difference
• Geometric Sequence: Each term is multiplied by a constant value
  • Formula: a_n = a_n-1 × r
  • Where r is the common ratio
• Second-Order Recursive Sequence: Each term depends on the two previous terms
  • Formula: a_n = c₁ × a_n-1 + c₂ × a_n-2
  • Where c₁ and c₂ are coefficients

The famous Fibonacci sequence (1, 1, 2, 3, 5, 8…) follows the recursive rule F_n = F_n-1 + F_n-2 with initial values F₁ = 1 and F₂ = 1. Each number is the sum of the two preceding ones, creating a pattern found throughout nature in phenomena like plant growth and shell spirals.

How to Find Recursive Rule?

• Select your sequence type from the dropdown menu: arithmetic, geometric, or custom recursive rule.
• For arithmetic sequences:
  • Enter the first term (a₁) – this is where your sequence starts
  • Enter the common difference (d) – the value added to each term
  • Specify how many terms you want to generate (default is 10)
  • Click “Calculate” to instantly see your sequence unfold
• For geometric sequences:
  • Enter the first term (a₁)
  • Enter the common ratio (r) – the value multiplied to each term
  • Click “Calculate” to see your exponential pattern
• For custom recursive formulas (like Fibonacci):
  • Enter the first two terms of your sequence
  • Specify the coefficients for the previous two terms
  • Click “Calculate” to generate complex patterns

Let’s track compound interest on $1,000 with 6% annual interest. This is a geometric sequence with first term a₁ = 1000 and common ratio r = 1.06. Using our calculator:

• Select “Geometric Sequence”
• Enter 1000 for First Term
• Enter 1.06 for Common Ratio
• Set Number of Terms to 10 for a 10-year projection
• Click “Calculate” to see your balance grow: $1,000, $1,060, $1,123.60…

The calculator provides both the recursive formula (a_n = a_n-1 × 1.06) and the explicit formula (a_n = 1000 × 1.06^n-1), plus all calculated terms with precise values.

What is Recursive Rule Formula?

A recursive rule formula is a mathematical expression that defines sequence terms by referring to previous terms in the same sequence. It serves as a set of instructions for generating the entire sequence from its beginning.

Example 1: College Tuition Increases (Arithmetic Sequence)

A college announces that this year’s tuition is $12,000 with planned annual increases of $500.

Recursive formula setup:

• First term: a₁ = $12,000
• Common difference: d = $500
• Recursive rule: a_n = a_n-1 + 500

Calculating the sequence:

• Year 1: a₁ = $12,000
• Year 2: a₂ = a₁ + $500 = $12,000 + $500 = $12,500
• Year 3: a₃ = a₂ + $500 = $12,500 + $500 = $13,000
• Year 4: a₄ = a₃ + $500 = $13,000 + $500 = $13,500

Using our calculator with these values generates the complete sequence, showing that tuition will reach $14,500 in year 6.

Example 2: Investment with Annual Returns (Geometric Sequence)

You invest $5,000 in a fund with 7% annual returns, reinvesting all gains.

Recursive formula setup:

• First term: a₁ = $5,000
• Common ratio: r = 1.07
• Recursive rule: a_n = a_n-1 × 1.07

Calculating the sequence:

• Year 1: a₁ = $5,000
• Year 2: a₂ = a₁ × 1.07 = $5,000 × 1.07 = $5,350
• Year 3: a₃ = a₂ × 1.07 = $5,350 × 1.07 = $5,724.50
• Year 4: a₄ = a₃ × 1.07 = $5,724.50 × 1.07 = $6,125.22

Our calculator instantly shows your investment growing to approximately $9,898 by year 10.

Example 3: Loan Repayment Planning (Custom Recursive Sequence)

You’re developing a custom loan repayment plan where each month’s payment is the average of the previous two months plus $25.

Recursive formula setup:

• First term: a₁ = $300 (first month’s payment)
• Second term: a₂ = $320 (second month’s payment)
• Coefficients: c₁ = 0.5, c₂ = 0.5 (for averaging), plus $25
• Recursive rule: a_n = 0.5 × a_n-1 + 0.5 × a_n-2 + 25

Calculating the sequence:

• Month 1: a₁ = $300
• Month 2: a₂ = $320
• Month 3: a₃ = 0.5 × a₂ + 0.5 × a₁ + 25 = 0.5 × $320 + 0.5 × $300 + $25 = $160 + $150 + $25 = $335
• Month 4: a₄ = 0.5 × a₃ + 0.5 × a₂ + 25 = 0.5 × $335 + 0.5 × $320 + $25 = $167.50 + $160 + $25 = $352.50

Using our calculator with these inputs shows the payment stabilizing around $377.50 per month over time.

References

• Strogatz, S. (2019). Infinite Powers: How Calculus Reveals the Secrets of the Universe. Houghton Mifflin Harcourt
• Khan Academy. (2023). Recursive formulas for sequences. Mathematical Thinking
• National Council of Teachers of Mathematics. (2022). Principles to Actions: Ensuring Mathematical Success for All. NCTM