Find a formula for the general term $a_{n}$ of the sequence, assuming that the pattern of the first few terms continues.

$\{1,0,-1,0,1,0,-1,0, \ldots\}$

To find: Formula for the general term an of the sequence.

The general term an of the sequence is sinnπ2 (or) cos(n−1)π2 .

Given:

The sequence is {1,0,−1,0,1,0,−1,0,...} .

Here, a1=1,a2=0,a3=−1,a4=0,a5=1,... .

Calculation:

The first term of the sequence can be written as a1=sinπ2 (or) cos 0 .

The second term of the sequence can be written as a2=sin2π2 or cosπ2 .

The third term of the sequence can be written as a3=sin3π2 or cos2π2 .

The fourth term of the sequence can be written as a4=sin4π2 (or) cos3π2 .

The fifth term of the sequence can be written as a5=sin5π2 (or) cos4π2 .

Proceed in a similar way and obtain the nth term of the sequence as an=sinnπ2 (or) cos(n−1)π2 .

Therefore, the formula for general term an of the sequence is sinnπ2 (or) cos(n−1)π2_ .