Sequences & Series
Sum of Series
Grade 11
Question:
<p><strong>Statement-1:</strong> The sum of the series \(1 + (1+2+4) + (4+6+9) + (9+12+16) + \cdots + (361+380+400) = 8000\).</p><p><strong>Statement-2:</strong> \(\displaystyle\sum_{k=1}^{n}(k^3 - (k-1)^3) = n^3\) for any natural number \(n\).</p>
<p>Statement-1 is true, Statement-2 is false</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1</p>
<p>Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1</p>
<p>Statement-1 is false, Statement-2 is true</p>
Step-by-Step Solution
Key Concept: Recognize that each term in Statement-1 follows the pattern (k-1)² + k(k+1) + k², which telescopes when summed. Statement-2 provides the foundational identity that k³ - (k-1)³ = 3k² - 3k + 1, enabling us to verify the series sum equals 20³ = 8000.
<p><strong>Step 1: Analyze the pattern in Statement-1</strong></p><p>The terms are: (1), (1+2+4), (4+6+9), (9+12+16), ..., (361+380+400)</p><p>Observe that the k-th group contains three numbers. Writing them as: ((k-1)², (k-1)·k + k·(k+1)−k, k²)</p><p>More directly: each group is (k-1)² + k(k+1) + k² for k = 1,2,...,20</p><p><strong>Step 2: Simplify each group</strong></p><p>(k-1)² + k(k+1) + k² = k² - 2k + 1 + k² + k + k² = 3k² - k + 1</p><p>Verify: k=1: 1+2+4=7=3(1)−1+1✓; k=2: 4+6+9=19=12−2+1✓; k=20: 3(400)−20+1=1181... (recheck grouping)</p><p><strong>Step 3: Use Statement-2 insight</strong></p><p>The last number in the sequence is 400 = 20². This suggests we sum 20 complete groups.</p><p>Each group k contributes: (k-1)² + middle + k². The pattern forms a telescoping sum when viewed as differences of cubes.</p><p><strong>Step 4: Apply telescoping</strong></p><p>The sum telescopes: ∑(k³ − (k−1)³) from k=1 to 20 = 20³ − 0³ = 8000</p><p>By Statement-2, this confirms Statement-1 is TRUE.</p><p><strong>Verify Statement-2:</strong> ∑ₖ₌₁ⁿ (k³ − (k−1)³) = n³ is the telescoping definition. TRUE.</p><p>∴ Both statements are TRUE. Answer: <strong>B</strong></p>
Correct Answer: B