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uysha [10]
3 years ago
14

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

Mathematics
1 answer:
sergiy2304 [10]3 years ago
8 0
                       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
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