Question

Which of the following series can be expressed as recursive or non-recursive sequences? Choose all that apply.

a) Arithmetic series
b) Geometric series
c) Fibonacci series
d) Harmonic series

Answer 1

Arithmetic, geometric, Fibonacci, and harmonic series can be expressed as recursive or non-recursive sequences, with Fibonacci being most commonly defined recursively. The other series are typically represented with non-recursive formulas.

Step-by-step explanation:

The question you've asked is about whether certain series can be expressed as recursive or non-recursive sequences. Here are the details for each:

• Arithmetic series: This can be expressed as a non-recursive sequence where each term is obtained by adding a constant to the previous term, represented as a_n = a_1 + (n-1)d, where 'd' is the common difference.
• Geometric series: Similarly to arithmetic series, a geometric series can also be expressed as a non-recursive sequence with the formula a_n = a_1 * r^(n-1), where 'r' is the common ratio.
• Fibonacci series: The Fibonacci series is a classic example of a recursive sequence where each term is the sum of the two preceding ones, usually starting with 0 and 1. It's defined by the recursive formula F_n = F_{n-1} + F_{n-2}.
• Harmonic series: This series can be expressed as a non-recursive sequence, with the nth term given by the formula 1/n.

All of the series mentioned above can be represented either recursively or non-recursively, however, the Fibonacci series is most commonly defined as a recursive sequence.