Some sequences refuse to give you a direct formula — instead they tell you how to get the next term from the current one. That's a recursive formula: a₁ = 3, aₙ = 2aₙ₋₁ + 1 generates 3, 7, 15, 31, 63, each term built from its predecessor. It's how nature actually computes: rabbit populations, compound interest month by month, and the Fibonacci sequence (each term the sum of the two before it) all unfold recursively. Computer programmers recognize it instantly — recursion is a cornerstone of algorithms, from sorting to tree traversal.

The trade-off is clear: recursion is easy to write and tedious to evaluate far out. Finding the 100th term by hand means computing 99 predecessors, which is why mathematicians hunt for closed forms (the explicit formula aₙ = 4·2ⁿ⁻¹ − 1 for the example above jumps straight to any term). This calculator evaluates recursive steps: enter the current term value and the recursion's parameters, and read the next term. Use it to generate the first several terms quickly, then look for the pattern — constant differences signal linear growth, constant ratios signal exponential, and second differences constant signal quadratic. Naming the pattern is the first step toward the closed form.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Recursive Formula Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.