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Write a formula for a sequence.

a) aₙ = aₙ₋₁ + aₙ₋₂
b) aₙ = n²
c) aₙ = 1/n
d) aₙ = 2ⁿ

User Elector
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1 Answer

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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.

User Titaniumdecoy
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