Question

Numerele naturale impare consecutive sunt grupate astea : {1}, {3,5} , {7,9,11} , {13,15,17,19},......etc
Determinati suma numerelor din a 8 a multime.

Answer 1

Răspuns:

{1}

{3,5}

{7,9,11}

{13, 15, 17, 19}

{21,23,25,27,29}

{31,33,35,37,39,41}

{43,45,47,49,51,53,55}

{57,59,61,63,65,67,69,71}

57+59+61+63+65+67+69+71=521

Answer 2

{1}, {3,5} , {7,9,11} , {13,15,17,19}, ...

Multimile sunt de forma:

{1}, {1,3,5} \ {1}, {1,3,5,7,9,11} \ {1,3,5}, {1,3,5,7,9,11,13,15,17,19} \ {1,3,5,7,9,11}, ...

Observăm că au tiparul:

$A_n =\bigcup\limits_{i=1}^{\frac{n(n+1)}{2}}\{2i-1\}\,\backslash\bigcup\limits_{i=1}^{\frac{n(n-1)}{2}}\{2i-1\}$

Calculăm a 8-a mulțime a șirului:

$A_8 =\bigcup\limits_{i=1}^{\frac{8(8+1)}{2}}\{2i-1\}\,\backslash\bigcup\limits_{i=1}^{\frac{8(8-1)}{2}}\{2i-1\}\\ \\ A_8 =\bigcup\limits_{i=1}^{36}\{2i-1\}\,\backslash\,\bigcup\limits_{i=1}^{28}\{2i-1\}\\ \\ A_8 = \{1,3,5,7,...,71\}\,\backslash\, \{1,3,5,7,...,55\} \\\\ A_8 = \{57,59,61,...,71\}$

Înseamnă că suma numerelor din a 8-a mulțime este:

$S = 57+59+61+...+77\\ S = 1+3+5+...+71 -(1+3+5+...+55)\\\\S = \Big(\dfrac{71+1}{2}\Big)^2 - \Big(\dfrac{55+1}{2}\Big)^2\\\\S = 36^2 - 28^2 \\ S = (36-28)(36+28) \\ S = 8\cdot 67\\ \\ \Rightarrow \boxed{S = 512}$