For α, β ∈ R and a natural number n, let A_r = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>r</mi></mtd><mtd><mn>1</mn></mtd><mtd><mfrac><msup><mi>n</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mi>α</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>r</mi></mtd><mtd><mn>2</mn></mtd><mtd><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mi>β</mi></mtd></mtr><mtr><mtd><mn>3</mn><mi>r</mi><mo>-</mo><mn>2</mn></mtd><mtd><mn>3</mn></mtd><mtd><mfrac><mrow><mi>n</mi><mo>(</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math>. Then 2A_10 - A_8 is
Step-by-Step Solution
Key Concept: Observe the columns of the determinant. Notice that the second column is a multiple of the first column if we consider the linear relationship between rows or columns. Specifically, check if the determinant is zero for any r.
Let the determinant be A_r. Observe the columns C1, C2, C3. Notice that the rows are linearly dependent. Specifically, R2 = 2*R1 is not true, but check the determinant value. For any r, the determinant A_r = 0 because the columns are linearly dependent. Since A_r = 0 for all r, 2A_10 - A_8 = 2(0) - 0 = 0.
Correct Answer: 4