Selena Cowan

2022-01-25

What's the difference between arithmetic, geometric, harmonic and Fabonacci sequences?

sjkuzy5

Expert

A sequence is called an arythmetic sequence if the difference of the term and the previous term is always the same.
For example, ${a}_{n+1}-{a}_{n}=cons\mathrm{tan}t\left(d\right)$
the constant difference, generally denoted by (d) is called the common difference.
Ex. $1,4,7,10,\dots$ is an arythmetic sequence, whose first term is $1$ and the common difference is equal to $\left(4-1=3\right)$
A sequence is called a geometric sequence if $\frac{{a}_{n+1}}{{a}_{n}}=cons\mathrm{tan}t$ for all n.
Ex. the sequence $4,12,36,\dots$ is a geometric sequence, because $\frac{12}{4}=\frac{36}{12}=\dots =3$, which is constant.
A sequence is harmonic when each term is harmonic mean of the neighboring terms.
For example, $\frac{1}{a},\frac{1}{a+d},\frac{1}{a+2d},\dots$, here $\left(a\ne 0\right)$
Ex. $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$
here, $a=1$ (first term)
$d=1$ (common difference)
The Fibonacci sequence starts like this: $0,1,1,2,3,5,8,13,21,\dots$
Each number is the sum of the two numbers that precede it.

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