Answer:
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 = [tex]\frac{N(N+1)(2N+1)}{6}[/tex]
Step-by-step 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 = [tex]\frac{N(N+1)(2N+1)}{6}[/tex]
