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📐 Mathematics  ·  Sequences and Series  ·  JEE

Sum: 1×2 + 2×3 + 3×4 + ... + n(n+1) =

Answer: n(n+1)(n+2)/3.

  • A n(n+1)(n+2)/3
  • B n(n+1)/2
  • C n²(n+1)²/4
  • D n(n+1)(2n+1)/6

Correct answer: A. n(n+1)(n+2)/3

Explanation: Σk(k+1) = Σk² + Σk = n(n+1)(2n+1)/6 + n(n+1)/2. Factor n(n+1): n(n+1)[(2n+1)/6 + 3/6] = n(n+1)(2n+4)/6 = n(n+1)(n+2)/3.

Each bar grows by the common difference d=3a1=2a2=5a3=8a4=11d=3

Bar heights for AP 2, 5, 8, 11 with the common difference d=3 marked between consecutive terms.

Concept context

Arithmetic progressions, geometric progressions, and sums

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