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Answer :
To find how many squares can be formed in a 25x25 grid, we need to consider squares of all possible sizes that can fit within the grid.
Here’s a step-by-step explanation of the process:
Understanding the Grid: The grid is a 25x25 grid, made up of 25 rows and 25 columns of small unit squares.
Calculating Squares of Different Sizes:
- A square of size 1x1 can fit in each individual cell of the grid. Since there are 25 rows and 25 columns, there are [tex]25 \times 25 = 625[/tex] such squares.
- Similarly, a square of size 2x2 can be formed by selecting a 2x2 sub-grid. The top-left corner of such a square can start from any of the first 24 rows and any of the first 24 columns, because the last row and column would extend the square outside of the grid. So, there are [tex]24 \times 24 = 576[/tex] 2x2 squares.
- For a square of size 3x3, it can start from any of the first 23 rows and 23 columns, so there are [tex]23 \times 23 = 529[/tex] such squares.
- This pattern continues until we consider a square of size 25x25, which can only start from the very first cell of the grid, giving [tex]1 \times 1 = 1[/tex] square of this size.
General Formula: For a square of size [tex]n \times n[/tex], there are [tex](25-n+1) \times (25-n+1)[/tex] such squares.
Summing All Possible Squares: You need to calculate the total number of squares by summing up all the squares of different sizes:
[tex]\sum_{n=1}^{25} (25-n+1)^2 = 1^2 + 2^2 + 3^2 + ... + 25^2[/tex]Using the Formula for Sum of Squares: The sum of the squares of the first [tex]N[/tex] natural numbers is given by:
[tex]\sum_{k=1}^{N} k^2 = \frac{N(N+1)(2N+1)}{6}[/tex]- Plug [tex]N = 25[/tex] into this formula:
[tex]\frac{25 \times 26 \times 51}{6} = 5525[/tex]
- Plug [tex]N = 25[/tex] into this formula:
Therefore, the total number of squares that can be formed in a 25x25 grid is 5525.
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