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$)$

$1$

C

$1$

is a checkerboard with $1$ black square.

Draw a separate $2$

C

$2$

from $1$

C

$1$

by adding only red and white squares.

Draw a separate $3$

C

$3$

from $2$

C

$2$

by adding only black and white squares.

Continue drawing separate $4$

C

$4$

and $5$

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 $1$

a

$1$

$,2$

a

$2$

$,3$

a

$3$

$,4$

a

$4$

$,5$

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 answ