Explicație pas cu pas:
5, 6, 10, 19, 35...
[tex]T_{1} = 5[/tex]
[tex]T_{2} = 6 = 5 + {1}^{2} \\ = > T_{2} = T_{1} + {1}^{2}[/tex]
[tex]T_{3} = 10 = 6 + {2}^{2} = 5 + {1}^{2} + {2}^{2} \\ = > T_{3} = T_{1} + {1}^{2} + {2}^{2}[/tex]
[tex]T_{4} = 19 = 10 + {3}^{2}[/tex]
[tex]= > T_{4} = T_{1} + {1}^{2} + {2}^{2} + {3}^{2}[/tex]
observăm că:
[tex]T_{k} = T_{1} + ({1}^{2} + {2}^{2} + {3}^{2} + ... + {(k - 1)}^{2}) \\ [/tex]
cunoaștem formula:
[tex]{1}^{2} + {2}^{2} + {3}^{2} + ... + {n}^{2} = \frac{n(n + 1)(2n + 1)}{6} \\ [/tex]
→
[tex]T_{n} = T_{1} + \frac{n(n - 1)(2n - 1)}{6} \\ [/tex]
a)
[tex]T_{7} = T_{1} + ({1}^{2} + {2}^{2} + {3}^{2} + ... + {6}^{2}) \\ = 5 + \frac{7 \times 6 \times 13}{6} = 5 + 91 = 96 \\ [/tex]
b)
[tex]T_{18} = 5 + \frac{18 \times 17 \times 35}{6} = 5 + 1785 = 1790\\[/tex]
[tex]T_{19} = 5 + \frac{19 \times 18 \times 37}{6} = 5 + 2109 = 2114\\ [/tex]
→
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