Question

Specifies the step by which each term of the sequence is generated from the preceding term or terms.

a) Recursive formula
b) Explicit formula
c) Arithmetic progression
d) Geometric progression

Answer 1

The question is about mathematical sequences, where a recursive formula uses previous terms, an explicit formula allows direct calculation, and arithmetic and geometric progressions are sequences with a constant difference or ratio, respectively.

Step-by-step explanation:

The student is asking about the methods of defining sequences in mathematics. Specifically:

• Recursive formula: A formula that defines the terms of a sequence by relating each term to its predecessor(s).
• Explicit formula: A formula that allows computation of any term in the sequence directly by plugging in the value of its position in the sequence.
• Arithmetic progression: A sequence in which each term after the first is obtained by adding a constant (common difference) to the previous term.
• Geometric progression: A sequence in which each term after the first is obtained by multiplying the previous term by a constant (common ratio).

Regarding series expansions, the Binomial theorem provides an explicit formula for the terms in the expansion of the binomial expression (a + b)n.

For sequences showing exponential growth, each term is generated by multiplying the preceding term by a constant factor, as exemplified in sequences where the term doubles at each step, described by 2n.