Each gives the first term or two of a sequence along with a...

Each gives the first term or two of a sequence along with a recursion formula for the remaining terms. Write out the first ten terms of the sequence.
$$a_{1}=2, \quad a_{2}=-1, \quad a_{n+2}=a_{n+1} / a_{n}$$

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Key Concepts

Initial Conditions

Initial conditions specify the starting values of a sequence and are crucial for uniquely determining the sequence when using a recurrence relation. Without these fixed starting points, the recurrence relation could yield multiple sequences.

Iterative Computation

Iterative computation is the process of repeatedly applying the recurrence relation, starting from the initial conditions, to calculate successive terms in the sequence. This method is essential for generating the explicit terms and revealing patterns or behaviors within the sequence.

Sequence

A sequence is an ordered list of elements, typically numbers, where each element is determined by a specific rule or pattern. In mathematical problems, sequences provide a foundation for exploring patterns, convergence, and various properties that arise from the way elements are defined.

Recurrence Relation

A recurrence relation is an equation that recursively defines a sequence: each term is given as a function of the preceding terms. This concept is fundamental in discrete mathematics and allows one to determine further elements in a sequence once initial values are provided.

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