<p>The set of natural number <em>N</em> is partitioned into arrays of rows and columns in the form of matrices as \(m_1 = [1], m_2 = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}, m_3 = \begin{bmatrix} 6 & 7 & 8 \\ 9 & 10 & 11 \\ 12 & 13 & 14 \end{bmatrix}, \cdots m_n = \begin{bmatrix} . & . & . \\ . & . & . \\ . & . & . \end{bmatrix}\) and so on, then sum of the elements of the diagonal in \(m_{10}\) is ___________.</p>
Step-by-Step Solution
Key Concept: Matrix m_n is an n×n matrix starting from the natural number that comes after all elements in m₁, m₂, ..., m_(n-1) are exhausted. Find the starting element of m₁₀ using the formula: total elements before m_n = 1² + 2² + ... + (n-1)², then sum the diagonal elements.
<p><strong>Step 1:</strong> Find the starting element of m₁₀.</p><p>Total elements in m₁, m₂, ..., m₉ = 1² + 2² + 3² + ... + 9²</p><p>Using formula: Σk² = n(n+1)(2n+1)/6</p><p>Sum = 9(10)(19)/6 = 1710/6 = 285</p><p>So m₁₀ starts at element 286.</p><p><strong>Step 2:</strong> Identify diagonal elements of m₁₀ (a 10×10 matrix).</p><p>m₁₀ has elements from 286 to 286 + 100 - 1 = 385</p><p>The diagonal elements are at positions (1,1), (2,2), (3,3), ..., (10,10)</p><p>In a row-wise filled matrix, element at position (i,i) = 286 + 10(i-1) + (i-1) = 286 + 11(i-1) = 275 + 11i</p><p><strong>Step 3:</strong> Sum the diagonal elements.</p><p>Sum = Σ(275 + 11i) for i = 1 to 10</p><p>= 275(10) + 11(1 + 2 + ... + 10)</p><p>= 2750 + 11(55)</p><p>= 2750 + 605 = 3355</p><p>∴ Answer: <strong>3355</strong></p>
Correct Answer: 3355