★ Must-know
📌 A sequence can be defined explicitly by a formula such as or recursively by giving initial term(s) and a relation connecting consecutive terms.
To calculate successive terms of a recursively defined sequence, start from the given initial term(s), substitute them into the recurrence, and repeat the relation until the requested index is reached.
A recurrence model translates a verbal evolution into algebra by identifying the previous value, the next value, and any multiplication, addition, percentage change, or averaging operation between them.
Further detail
Explicit formula gives a term directly, whereas recurrence builds it from previous terms.
1. Which statement correctly distinguishes an explicit definition from a recursive definition of a sequence?
2. A sequence is defined by and . What is ?
3. Which condition characterizes an arithmetic sequence?
How can a sequence be defined explicitly?
By a formula like .
What is the first step to calculate terms of a recursive sequence?
Start from the given initial term(s).
What defines an arithmetic sequence?
Adding the same number called common difference to each term gives the next term.
What is the explicit formula for an arithmetic sequence with first term and difference ?
for every natural number .
Sequence notation label
Explicit or recursive definition
Arithmetic sequence formula
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