The table below shows a flooring consisting of square tiles measuring 1 m^2 each. Each day similar tiles are added to the previous pattern.

1. Find the $n$th term of the sequence for the area added.

The areas added are $1, 5, 9, \ldots$

This is an arithmetic sequence with common difference $4$.

$\text{Area on Day }n = 1 + (n-1)\times 4$

$= 1 + 4n - 4 = 4n - 3$

1. Evaluate the Day $n$ expression when $n=20$.

From part (a)(i), area added on Day $n$ is $4n-3$.

For Day 20,
$4(20)-3=80-3=77$.

1. Rewrite the expression for the area added in a form that shows it is odd.

From part (a)(i), the area added on Day $n$ is
$4n-3$.

$4n-3=4n-4+1=4(n-1)+1=2[2(n-1)]+1$.

This is of the form $2k+1$, so it is always odd.

1. Add the area added from Day 1 to Day 6 to get the total area on Day 6.

Using the sequence of areas added: $1, 5, 9, 13, 17, 21$.

Total area on Day 6
$=1+5+9+13+17+21=66$

1. Write the total area as the sum of the areas added from Day 1 to Day $n$.

From part (a)(i), the area added on Day $r$ is $4r-3$. Hence
$\text{Total area}=\sum_{r=1}^{n}(4r-3).$

2. Sum and simplify into the required form $an^2+bn$.

$\sum_{r=1}^{n}(4r-3)=4\sum_{r=1}^{n}r-3n$
$=4\left(\frac{n(n+1)}{2}\right)-3n$
$=2n(n+1)-3n$
$=2n^2+2n-3n$
$=2n^2-n.$

1. Use the total area formula to find when the total area is 780 m².

From part (b)(ii), total area on Day $n$ is
$2n^2-n.$
Set this equal to $780$:
$2n^2-n=780.$
Checking $n=20$,
$2(20)^2-20=800-20=780.$

2. Compare the required number of days with 3 weeks.

$3$ weeks $=21$ days.

Since Day $20$ is within $21$ days, the flooring can be completed in $3$ weeks.