Selena Cowan

Answered

2022-01-25

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

Answer & Explanation

sjkuzy5

Expert

2022-01-26Added 11 answers

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 $(4-1=3)$

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 $(a\ne 0)$

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.

For example,

the constant difference, generally denoted by (d) is called the common difference.

Ex.

A sequence is called a geometric sequence if

Ex. the sequence

A sequence is harmonic when each term is harmonic mean of the neighboring terms.

For example,

Ex.

here,

The Fibonacci sequence starts like this:

Each number is the sum of the two numbers that precede it.

Jaiden Conrad

Expert

2022-01-27Added 14 answers

Arithmetic sequence is a sequence where the difference between two consecutive terms is constant while a geometric sequence is a sequence where the ratio between two consecutive terms is constant.

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