Special Checkerboard $-$ Part $2$ of $3$: Iteration

Continuing the previous question:

You will now use iteration to deduce a partial solution involving $\backslash$Sigma

or $\backslash$Pi

operators for the sequence $$

:

Give the first $5$ terms of the sequence. Show and keep the intermediate expansions because they are more important than the final values for noticing a pattern $($and your grade will depend on it$).$

Guess a non$-$recursive formula which describes the sequence. The formula should include $\backslash$Sigma

or $\backslash$Pi

operators and should be as compact as possible.

The pedagogical goal of this question is not to find an analytical solution for $$

$,$ but to learn how to use iteration to notice patterns in sequences, and to write them correctly and succinctly using $\backslash$Sigma

and $\backslash$Pi

notation.

In order to do this, you must work from intermediate values instead of final values. Do distribute your operations to remove the parentheses in each term of the sequence, but do not calculate the results of additions, multiplications, and exponentiations, because if you do the pattern will disappear.

This problem is continued in the next question.