Induction Problem Number of Tiles on Floor

Question

I took a discrete math course about a year ago, and I recently decided to crack open my book again as a refresher on induction proofs and problems. I ran across this problem, which I didn't remember solving, and it is currently giving me quite a bit of trouble. I am mainly struggling with part d, where it asks me to find a formula for the total number of tiles placed on the floor. I know the number of squares on the floor for a given k will be equal to k squared, and that each strip will have (11 - k)(2k - 1) tiles. I need some help or hints trying to find a formula that, for any k, I can find the total number of tiles on the floor.

Answer 1

As you said each strip $S_k$ has (11-k)(2k-1) tiles. You are being asked to find the total number of tiles on the 10 strips which should be:

$\sum_{k=1}^{10} S_k = \sum_{k=1}^{10} (11-k)(2k-1)$
$\quad\quad\quad\quad = \sum_{k=1}^{10} 23k-2k^2-11$
$\quad\quad\quad\quad = 23\sum_{k=1}^{10} k-2\sum_{k=1}^{10}k^2-\sum_{k=1}^{10}11$

and you know:
$\sum_{k=1}^{n} k= \frac{(n)(n+1)}{2}$

$\sum_{k=1}^{n} k^2= \frac{(n)(n+1)(2n+1)}{6}$ (which is formula 7 in your link)

$\sum_{k=1}^{10}11 = 110$