Q$2(5$ points$)$

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

Continuing the previous question:

You will now use iteration to deduce a partial solution involving $$ or $$ operators for the sequence $a_n$ :

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 $$ or $$ operators and should be as compact as

possible.

The pedagogical goal of this question is not to find an analytical solution for $a_n,$ but to learn how to use iteration to

notice patterns in sequences, and to write them correctly and succinctly using $$ and $$ 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.Q$3(5$ points$)$

Special Checkerboard $-$ Part $3$ of $3$: Guess an analytical solution

You will now turn these checkerboards into two$-$tone checkerboards, where the white squares are untouched, but some of the

dark squares will now be red instead of black. $($Any colour other than white is considered dark for this exercise$)$

$C_1$ is a checkerboard with $1$ black square.

Draw a separate $C_2$ from $C_1$ by adding only red and white squares.

Draw a separate $C_3$ from $C_2$ by adding only black and white squares.

Continue drawing separate $C_4$ and $C_5,$ each time alternating between the colours red and black. You are asked to draw separate

checkerboards because this will help you see the pattern much better.

Notice that $a_n=$ number of black squares in $C_n+$ number of red squares in $C_n.$

Look at the checkerboards you have just drawn, and express each of $a_1,a_2,a_3,a_4,a_5$ as a sum of two numbers using the colour

coding of the drawings. Based on these values, guess a non$-$recursive formula for $a_n.$ Explain your answer

Note that there is more than one way to draw the two$-$tone checkerboards gradually as described in this question. However, only

one pattern will give you an obvious formula. If you are not finding this formula, try building the checkerboards differently.