Using Technology for Recurrence Relations

Why Use Technology?

While simple sequences can be calculated by hand, financial problems often require values after 10, 20, or 30+ periods. Technology makes this fast, accurate, and practical.

CAS Calculator: Generating a Sequence

Most VCAA-approved CAS calculators (TI-Nspire, Casio ClassPad) can generate sequences using:

Method 1: Recursive/List Mode

Enter the recurrence relation directly:
- TI-Nspire: Use the Sequence function or a list with formula referencing the previous term
- Casio ClassPad: Use Sequence mode or Recursion table

Method 2: Spreadsheet-Style Entry

Enter $V_0$ in one cell, then use a formula in the next cell referencing the previous. Copy down.

Method 3: Financial Solver

For financial applications, use the TVM (Time Value of Money) solver:
- N = number of periods
- I% = interest rate per period
- PV = present value (starting balance)
- PMT = payment each period (negative if paying out)
- FV = future value

Worked Example: Using TVM Solver

Problem: \$5000 invested at 6% p.a. compound interest. Find value after 8 years.

TVM inputs:
- $N = 8$
- $I\% = 6$
- $PV = -5000$ (negative: money invested)
- $PMT = 0$
- Solve for $FV$

Output: $FV = 7969.24$

Verification by recurrence: $V_{n+1} = 1.06 \times V_n, \quad V_0 = 5000$

Reading Sequence Tables

Technology displays sequences in a table. Key skills:

• Identify when a balance first exceeds/falls below a value: scan the table column
• Find total interest paid: final balance − initial balance (for investments)
• Find when a loan is paid off: find when $V_n \leq 0$

Generating a Graph

Plotting $V_n$ vs $n$ shows the pattern:
- Arithmetic: straight line
- Geometric (growth): upward curve
- Loan/annuity: downward curve toward zero

Checking Your Recurrence Relation with Technology

1. Enter the recurrence relation into your CAS
2. Generate the first few terms
3. Check they match hand calculations for $V_1$ and $V_2$
4. If they don’t match, re-examine your $R$ and $d$

KEY TAKEAWAY: Technology lets you find any term in a sequence quickly and check your work. Always verify $V_1$ by hand to confirm your recurrence relation is correct before using technology for later terms.

EXAM TIP: In VCAA exams, you may be given a table of values from a recurrence relation and asked to identify the rule, find a missing value, or determine when a condition is first met. Practice reading these tables efficiently.

VCAA FOCUS: You are expected to use technology effectively. Know how to enter recurrence relations and use the TVM solver on your approved calculator. Show key values from the technology output in your working.