Explicit or Recursive: The Sequence Staircase
About this app
An arithmetic sequence as a staircase of base a and blocks d: recursively step by step (asking down to step 1, then hopping up, one moon per hop) vs. explicitly aₙ = a + (n−1)·d. Up to step n there are n − 1 hops; a ghost block refutes "a + n·d".
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Two formulas for the same sequence
An arithmetic sequence always grows by the same amount d. Recursively you write aₙ = aₙ₋₁ + d with the start value a₁ = a: each term comes from the one before it. Explicitly you write aₙ = a + (n − 1) · d: you plug in the index n and get the term at once. Both formulas describe the same sequence, for example output that is a tonnes in the first year and rises by d tonnes every year.
What the app shows
The sequence stands as a staircase. Each column has the base a at the bottom and one block d on top for every step; step 1 is only the base. On the recursive way a question chain runs from the target step down to step 1, then a ball hops up step by step, with a moon above each hop. On the explicit way the blocks above the base are counted. Up to step 4 there are three blocks and three hops, not four.
Try it yourself
Set the base a, the step height d and the number of steps, then tap the column you want. Compare the recursive way for step 4 and for step 18: recursively the effort grows with n, explicitly it stays a single substitution. The button "Mistake a + n·d" shows with a ghost block where the typical counting error leads. The Formula panel works out both ways with numbers.
Frequently asked questions
- What is the difference between the explicit and the recursive form of a sequence?
- The recursive form describes a term through its predecessor, for an arithmetic sequence aₙ = aₙ₋₁ + d with a₁ = a. The explicit form gives the term directly as a formula in n, here aₙ = a + (n − 1) · d. Recursively you need all earlier terms; explicitly it is enough to plug in n.
- Why does the formula of an arithmetic sequence contain n − 1?
- The first term is already a; no d has been added yet. Up to the n-th term you therefore take only n − 1 steps. It is like counting nights: from day 1 to day 4 you sleep three times. Computing a + n · d lands one term too high.
- What is the recursive formula of a geometric sequence?
- In a geometric sequence you multiply by a fixed factor q instead of adding a fixed amount: aₙ = q · aₙ₋₁ with a₁ = a. The explicit form is aₙ = a · qⁿ⁻¹, again with n − 1 in the exponent, because the first term a is not yet multiplied.
- What can I set in the app?
- You set the base a, the step height d, the number of steps shown and the step n you are looking for. Two buttons play the recursive and the explicit way, a third shows the mistake a + n · d. The Formula panel computes both ways with numbers.
Subject: Finance
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