In a $7\times 7$ square grid, how many squares can be formed in total ?

To find the total number of squares in an $n\times n$ grid, we sum the squares of consecutive natural numbers up to $n$:
$$\text{Total Squares}=\sum _{k=1}^n{k}^2=1^2+2^2+3^2+\cdots +n^2$$

For a $7\times 7$ grid ($n=7$):

$$\text{Total Squares}=1^2+2^2+3^2+4^2+5^2+6^2+7^2$$

$$\text{Total Squares}=1+4+9+16+25+36+49=140$$

Alternatively, using the formula $\frac{n(n+1)(2n+1)}{6}$:

$$\text{Total Squares}=\frac{7\times (7+1)\times (2(7)+1)}{6}=\frac{7\times 8\times 15}{6}=\frac{840}{6}=140$$