Question

Determine if the following sequence is arithmetic, geometric, or neither. s, t, 2t-s

Answer 1

Arithmetic sequence. Common difference: (t - s)

Step-by-step explanation

The terms for a geometric sequence has this form:

`a_n=ar^((n-1))`

So a = a1. In this case a = s. If this sequence were geometric, the second term should involve s. Therefore, it's not geometric.

The terms for an arithmetic sequence are:

`a_n=a+(n-1)d`

The first term is also a = a1. In tihs case, the first term is s, so a = s

Assuming this is an arithmetic sequence, a2 = t. With this we can look for d, the common difference:

`\begin{gathered} t=s+(2-1)d \\ t=s+d \\ d=t-s \end{gathered}`

If d = (t-s), then the third term is:

`\begin{gathered} a_3=s+(3-1)(t-s) \\ a_3=s+2(t-2) \\ a_3=s+2t-2s \\ a_3=2t-s \end{gathered}`

Which is the same as the 3rd term of the given sequence. Therefore this is an arithmetic sequence and the common difference is (t - s)