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.