Write the first five terms of a sequence that is both arithmetic and geometric

Question

We are working on sequences in math and I'm just a bit confused on how to do this we did not get very much explained. 

Answer 1

An Arithmetic Sequence has terms with a "common difference," called d in most equations.  The value of d is added to the current term to determine the next term.

 

A Geometric Sequence has terms with a "common ratio," called r in most equations.  The value of r is multiplied by the current term to determine the next term.

 

Now, what values could we use to add to the current term and to multiply by the current term to get the next term using the same sequence??   Of course, we could add 0 (so, d=0) and multiply by 1 (so, r=1).

 

An example of the first five terms of such a sequence is:

   2, 2, 2, 2, 2

Answer 2

I don't believe there is a sequence that has 5 terms that is both arithmetic and geometric.

Perhaps the question was to give 5 terms for each type by themselves?

 

Sequences are a list of numbers that are identified by their order in the sequence,

the individual items are called "terms" and they are numbered "1st term", "2nd term", etc.

So we often use "n" to stand for the term number (instead of "x" in other cases).

 

Arithmetic sequences change by the same amount from one term to the next.

This is the same as a linear equation.

The terms can be given by the equation:

    S_n = m*n + b

(which looks like the y=mx+b of a line.)

So an example is S_n= 2*n + 3, we get the sequence of values:

  2*1+3, 2*2+3, 2*3+3, etc.

= 5, 7, 9, 11,13, 15, etc.

 

Geometric sequences have a constant _factor_ (multiplication) or ratio between the terms:

  S_n = S₁(r)^(n-1)

Everytime n gets 1 bigger the value gets an extra factor of "r".

When n=1, the first term, (r)^(n-1) is 1 so S₁ is the first term as we'd want (sometimes "a" is used instead).

An example is

  Sn = 5*2^(n-1)

  = 5, 10, 20, 40, 80, etc

 

Hope this helps.