<p><em>m</em> equispaced horizontal lines are intersected by <em>n</em> equispaced vertical lines. If the distance between two successive horizontal lines is same as that between two successive vertical lines, then find the number of squares formed by the lines if \(m < n\).</p>
Step-by-Step Solution
Key Concept: Squares of different sizes can be formed on the grid; a k×k square requires (m-k+1) positions horizontally and (n-k+1) positions vertically, so total squares = Σ(m-k+1)(n-k+1) for k=1 to m. Substituting r=m-k+1 transforms this to the given answer.
<p><strong>Step 1:</strong> Recognize that on an (m-1)×(n-1) grid of rectangles (formed by m horizontal and n vertical lines), we can form squares of various sizes: 1×1, 2×2, 3×3, ..., up to m×m.</p><p><strong>Step 2:</strong> For a k×k square, there are (m-k+1) possible horizontal positions and (n-k+1) possible vertical positions. So the number of k×k squares is (m-k+1)(n-k+1).</p><p><strong>Step 3:</strong> Total number of squares = Σ<sub>k=1</sub><sup>m</sup> (m-k+1)(n-k+1).</p><p><strong>Step 4:</strong> Substitute r = m-k+1, so when k=1, r=m and when k=m, r=1. This gives: Σ<sub>r=1</sub><sup>m-1</sup> r(n-m+r) = Σ<sub>r=1</sub><sup>m-1</sup> (m-r)(n-r).</p><p>∴ Answer: <span>Σ</span><sub>r=1</sub><sup>m-1</sup>(m-r)(n-r)</p>
Correct Answer: \(\sum_{r=1}^{m-1}(m-r)(n-r)\)