lesson
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If you are on step 50 of a skyscraper stairwell, you do not need to know where the ground floor is to reach step 51. You only need to know where you are standing right now and how many steps to take next.
That is the core idea of a recursive process: defining the next term in a sequence using the term that came immediately before it.
📊A clean horizontal sequence diagram showing numbered stepping stones labeled a₁, a₂, a₃, a₄, with values 5, 8, 11, 14. Curved animated arrows leap from 5 to 8 (+3), 8 to 11 (+3), and 11 to 14 (+3). Above the stones, a glowing highlight shows 'Current Term: aₙ' and points back to 'Previous Term: aₙ₋₁' with a label '+ d (Common Difference)'. Minimalist white card style (#ffffff) on light gray (#f8f9fa), border (#e6e6e6), text (#1e2945), blue accent (#22b7ff). Responsive layout under 350px width.
How do we write down this 'step-by-step' rule using precise mathematical symbols?
Decoding Sequence Subscripts
A sequence is an ordered list of numbers where each number is called a term. We use subscripts to label the position of each term, like an address.
The symbol an represents the current term at position n, while an−1 represents the previous term at position n−1.
📊A visual breakdown comparing index notation. Center card showing large text 'aₙ₋₁' with an arrow pointing to the subscript 'n - 1' labeled 'Position / Index (1 step back)', next to 'aₙ' with subscript 'n' labeled 'Current Position'. A warning box below shows: 'aₙ₋₁ is NOT the same as aₙ - 1' with a clear visual contrast (changing position index vs subtracting 1 from value). Clean modern styling with blue (#22b7ff) highlights and coral (#ff5c5c) warning accents.
Now that we can refer to consecutive terms, how do we turn an arithmetic pattern into an official formula?
The Two-Part Recursive Formula
In an arithmetic sequence, the difference between consecutive terms is always constant. We call this constant value the common difference (d).
Because every term comes from adding d to the term before it, a complete recursive formula always requires two parts: an initial starting term and an update rule.
📊A structured blueprint box displaying the general recursive formula: Line 1: 'a₁ = k (Initial Condition: where the sequence begins)', Line 2: 'aₙ = aₙ₋₁ + d for n ≥ 2 (Recursive Step: how to get the next term)'. Callouts point to 'a₁' (Starting Value), 'aₙ₋₁' (Previous Value), and '+ d' (Add Constant Difference). Dark navy text (#1e2945), sky blue container (#e0f2fe), border (#bae6fd), rounded 12px card, clean typography.