Question

what is the difference of the following arithmetic sequence 1/2, 3/4, 1, 5/4

Answer 1

First off, let's define the term, \(t_n\).
\(t_n\) means the \(n\)th term in a sequence so in the case of the sequence \(\{1,5,6,10\}\), \(t_1=1\), \(t_2=5\), \(t_3=6\) and so on and so forth.

In an arithmetic sequence, the common difference is a number between different terms such that:
\[t_{n+1}-t_n=t_{n+2}-t_{n+1}=t_{n+3}-t_{n+2}...\]
So in our case, we know:
\[\eqalign{
&t_1=1/2=2/4 \\
&t_2=3/4 \\
&t_3=1=4/4 \\
&t_4=5/4
}\]
So then according to our formula,
\[(t_2-t_1)=(t_3-t_2)=(t_4-t_3)\]
So by substituting our values:
\[\left(\frac{3}{4}-\frac{2}{4}\right)=\left(\frac{4}{4}-\frac{3}{4}\right)=\left(\frac{5}{4}-\frac{4}{4}\right)=\frac{1}{4}\]