Question

Help solving this math problem a mathematical explanation is needed answering the question

Answer 1

1. 204 squares

2.
`\displaystyle S_n=\sum_(i=1)^(i=n)i^2=(n(n+1)(2n+1))/(6)`

3.
`\displaystyle S_(10)=385`

`\displaystyle S_(100)=338,350`

`\displaystyle S_(N)=(N(N+1)(2N+1))/(6)`

Explanation:

Counting And Adding

Counting is a normal activity done by humans and (supposedly) even by animals. But when the count tends to be a repetitive and long proccess, we try to look for formulas to shorten our work. That is where summation comes in handy.

1. Mr. Reihl's floor looks like a 8x8 chess board, half of the tiles are black and half are white. We need to count how many squares are in that floor, including overlapping and bigger squares.

The first and most obvious count comes when each square is of 1x1 size. There are 8 rows and 8 columns of each, so there are 8^2 1x1 squares

When we count all the 2x2 squares, we find a total of 7 different (overlapping) squares in the first row (2 rows high). If we move one square down, we find another row of 7 different overlapping squares. We can move down until we find the 7th row. We have counted 7^2 2x2 different squares.

Let's take 3x3 squares. Following a similar procedure as before, we can find 6x6 of them, and so on until we find our giant 8x8 single square.

How many squares have we count? It's represented by the sum of squares from 1 to 8:

`S=1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2`

By manually adding all of them, we find S=204 different squares

2. We have already shown a pattern for the sum. We asked ourselves: will there be a formula to find that sum automatically (go figure if there is a 100x100 tile floor!)? The answer is yes, we can compute the sum of n squares with the formula .

`\displaystyle S_n=\sum_(i=1)^(i=n)i^2=(n(n+1)(2n+1))/(6)`

If we apply the formula for n=8, we get

`\displaystyle S_8=\sum_(i=1)^(i=8)i^2=(8(8+1)(2(8)+1))/(6)=(1224)/(6)=204`

The sum was found to be 204, as it did before. It allows us to generalize our results .

3. If n=10

`\displaystyle S_(10)=(10(11)(21))/(6)=(2310)/(6)=385`

If n=100

`\displaystyle S_(100)=(100(101)(201))/(6)=(2,030,100)/(6)=338,350`

Finally, if n=N (any integer positive number)

`\displaystyle S_(N)=(N(N+1)(2N+1))/(6)`