1²+3²+4²+.......+(2n-2)²= ⅓n(2n-1)(2n+1)

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28-02-2016
Mathematics

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1²+3²+4²+.......+(2n-2)²= ⅓n(2n-1)(2n+1)

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Panoyin Panoyin · Answer 1 · 2016-02-28T17:50:14+01:00

1² + 3² + 4² + 4(n - 1)² = ¹/₃n(2n - 1)(2n + 1)
1² + 3² + 4² + (2n - 2)² = ¹/₃n(2n - 1)(2n + 1)
  1 + 9 + 16 + (2n - 2)(2n - 2) = ¹/₃n(2n(2n + 1) - 1(2n + 1))
10 + 16 + (2n(2n - 2) - 2(2n - 2)) = ¹/₃n(2n(2n) + 2n(1) - 1(2n) - 1(1) 16 + (2n(2n) - 2n(2) - 2(2n) + 2(2)) = ¹/₃n(4n² + 2n - 2n - 1)
26 + (4n² - 4n - 4n + 4) = ¹/₃n(4n² - 1)
26 + (4n² - 8n + 4) = ¹/₃n(4n² - 1)
  26 + 4n² - 8n + 4 = ¹/₃n(4n²) - ¹/₃n(1)
  4n² - 8n + 4 + 26 = 1¹/₃n³ - ¹/₃n
  4n² - 8n + 30 = 1¹/₃n³ - ¹/₃n
  + ¹/₃n + ¹/₃n
   4n² - 7²/₃n + 30 = 1¹/₃n³
-1¹/₃n³ + 4n² - 7²/₃n + 30 = 0
-3(-1¹/₃n³ + 4n² - 7²/₃n + 30) = -3(0)
-3(-1¹/₃n³) - 3(4n²) - 3(-7²/₃n) - 3(30) = 0
  4n³ - 12n² + 23n - 90 = 0