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KVPY2015MathematicsSequences and Series

The largest perfect square that divides 2014³-2013³+2012³-2011³+ .+2³-1³ is

Options

  1. A1²
  2. B2²
  3. C1007²
  4. D2014²

Correct answer

C. 1007²

Step-by-step solution

S=2014³-2013³+2012³-2011³+ .+2³-1³ _ t=1 ²⁰¹⁴ r²+ _ t=1 ¹⁰⁷(2 r-1)(2 r) S = _ r=1 ²⁰¹⁴ r²+ _ r=1 ¹⁰⁰⁷ (4 r²-2 r ) S = (2014)(2015)(4029) 6 + 4 1007 1008 2015 6 - 2 1007 1008 2 =(1007)² 4031

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