Instead of a formula that jumps straight to any term, a recurrence relation says "to get the next term, do this to the current one." You build the sequence step by step, like a chain. Each term unlocks the next.
You always need the starting value t₁, the rule alone isn't enough.
Maria's solar account: tₙ₊₁ = 1.05·tₙ + 100, with t₁ = 2000. Find the first few terms.
On your Casio, type the rule once, then press = repeatedly (the Ans key) to spit out term after term.
• Always state the starting value t₁, the recurrence is incomplete without it.
• Do the multiply then the add/subtract, in that order, each step.
• Writing one from context: r = what's multiplied (1 + the rate), d = what's added (deposit) or subtracted (repayment).
• A subtraction (− d) means money leaving, a loan repayment or withdrawal.
Same rule applied over and over: multiply by 1.05, then add 100. Each result feeds into the next, that's a recurrence.
| Rule looks like | Type |
|---|---|
| tₙ₊₁ = tₙ + d | arithmetic (adding only) |
| tₙ₊₁ = r·tₙ | geometric (multiplying only) |
| tₙ₊₁ = r·tₙ ± d | combined (finance: loans & savings) |
Spot whether there's a multiply, an add/subtract, or both, that tells you the type.
Don't read yet, just have a go in your head:
t₂ = 1.05 × 2000 + 100 = 2100 + 100 = 2200.
t₃ = 1.05 × 2200 + 100 = ? + 100 = ?
A sequence is defined by tₙ₊₁ = tₙ + 8, t₁ = 15. Find t₅ (build it step by step). Check below.
A loan of $20,000 is charged 5% interest each year, and $2,000 is repaid each year. Write the recurrence relation.
In one sentence, out loud: what two ingredients make a complete recurrence relation?