Question

Help solving this math problem with a mathematical explanation about the answer

Answer 1

1) 1²+2²+3²+4²+5²+6²+7²+8² = 204

2) here the pattern is squares form with only squares of natural number of tiles i.e) 1,4,9.....

3) floor with 10 tiles long, the number of squares = 385

floor with 100 tiles long, the number of squares =338350

floor with N tiles long, the number of squares = ∑ N² =
`(N(N+1)(2N+1))/(6)`

Explanation:

We can make square only using number of tiles which are squares of natural numbers.

so, we need 1, 4, 9, 16 ..... number of tiles to form a square.

Now, if there are N number of squares along one side,

choosing 1 square at a time , we get N² number of squares

choosing 2 square at a time , we get (N-1)² number of squares

and so on...

choosing N square at a time , we get 1² number of squares

so the total number of squares = 1² + 2² +......+ N² = ∑N²

And ∑N² =
`(N(N+1)(2N+1))/(6)`