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.