Final answer:
The student's question involved writing formulas for various types of sequences. The answers include a recursive sequence (an = an-1 + an-2), a quadratic sequence (an = n2), a harmonic sequence (an = 1/n), and an exponential sequence (an = 2n). These formulas represent different mathematical patterns.
Step-by-step explanation:
The question asks for the formulas of sequences in various forms. Here's the explanation for each:
- an = an-1 + an-2 represents a recursive sequence where each term is the sum of the two preceding terms, typical of the Fibonacci sequence.
- an = n2 describes a sequence where each term is the square of its position in the sequence, e.g., 1, 4, 9, 16, ...
- an = 1/n indicates a harmonic sequence where each term is the reciprocal of its position in the sequence, e.g., 1, 1/2, 1/3, 1/4, ...
- an = 2n denotes a geometric sequence where each term is 2 raised to the power of its position in the sequence, showing exponential growth, e.g., 2, 4, 8, 16, ...
All these sequences have distinctly different patterns and can represent a range of mathematical concepts, from growth and decay to repetitive relationships.