Let me clarify. I only have the arithmetic mean. I don't have the data from which it was determined.
It is not possible to answer this question without information on whether the terms are of an arithmetic or geometric (or other) progression, and what the starting term is.
Sure. There are other applications of arithmetic, but algebra without arithmetic is impossible. A broad knowledge of arithmetic is essential for mastery of algebra.
Of course. Formulas are not encountered until the student progresses from arithmetic to algebra.
axioms or postulates
An axiom.
"Arithmetic mean" means the same as average. "Mean", without qualifiers, usually refers to the arithmetic mean. However, there are other types of "means", for example, the geometric mean.
It is not possible to answer this question without information on whether the terms are of an arithmetic or geometric (or other) progression, and what the starting term is.
Sure. There are other applications of arithmetic, but algebra without arithmetic is impossible. A broad knowledge of arithmetic is essential for mastery of algebra.
Of course. Formulas are not encountered until the student progresses from arithmetic to algebra.
uh... yes without a working memory you would not know (remember) how to do arithmetic
Edith M. Joy has written: 'Arithmetic without a pencil' -- subject(s): Mental arithmetic
Of course! People were doing real arithmetic long before the first computer!
sketch
There are several methods. Euler could find the value of a logarithm by doing a search using the geometric mean, arithmetic mean and a table. There is a series using that natural log that can be converted into base 10. I have my own method which is in the "related links" section.
axioms or postulates
The alphabet of every language can be used as a geometric tool. Without writing, geometry is nearly impossible.
An axiom.