Remarks for AP

Once the first term a and the common difference \(d\) are given, then the entire arithmetic
sequence can be generated. We say that an arithmetic sequence is completely determined by the
values of its first term a and its common difference \(d\) . This leads us to the good habit of writing
an opening sentence that reads “Let \(a\) be the first term, and \(d\) the common difference of the AP”
to define the arithmetic sequence in question.

To test whether a given sequence \(\{u_n\}\) is an A.P., one just need to check that the difference
\(u_n - u_{n-1}\) is a constant, independent of \(n\) . This test is often known as A.P. test. Because the difference between consecutive terms is always the same, the A.P. test can also be carried out
using instead the difference \(u_{n+1} - u_n\).