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General Term of Recursive Sequence Calculator

Online closed-form calculator: construct a geometric sequence from a_n=c·a_(n−1)+b to get the general term; c=1 is arithmetic.

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a_n=c^(n−1)a₁+b(1−c^(n−1))/(1−c)
Initial
c=1 arithmetic
Constant
Positive integer

📖 Tutorial | Closed Form

1. Definition
For c≠1, constructing a geometric sequence (fixed point) gives a_n=c^(n−1)a₁+b(1−c^(n−1))/(1−c).
2. Formula & Notation
a₁Initial value
3. Properties & Laws
  • Closed form for c≠1;
  • c=1 gives a₁+(n−1)b (arithmetic);
  • Fixed point b/(1−c);
  • Matches iteration.
4. Steps
  1. Enter a₁, c, b, n;
  2. Click "Find Term";
  3. Or click "Load Example".
5. Worked Examples
Example: a_n=2a_(n−1)+1 gives a_n=2^n−1, term 5 = 31.
6. Common Pitfalls
c=1 needs the arithmetic case;
Exponent n−1;
Mind the (1−c) sign.

❓ FAQ | Closed Form

Form?
c^(n−1)a₁+b(1−c^(n−1))/(1−c).
c=1?
Arithmetic.
Derive?
Fixed point / geometric.
Fixed point?
b/(1−c).
Vs iteration?
Same.
Why?
Direct term without looping.