Question

fie sirul de nr naturale 1;2*3;4*5*6;7*8*9*10;...sa se determine al 7- lea si al 100- lea termen​

Answer 1

1; 2·3; 4·5·6; 7·8·9·10; ...

$\dfrac{1}{1};\dfrac{1\cdot 2\cdot 3}{1}; \dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6}{1\cdot 2\cdot 3}; \dfrac{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10}{1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6}; ...$

$\Rightarrow a_n = \dfrac{\frac{n(n+1)}{2}!}{\frac{(n-1)n}{2}!}$

$a_7 = \dfrac{\frac{7\cdot 8}{2}!}{\frac{6\cdot 7}{2}!} = \dfrac{(7\cdot 4)!}{(3\cdot 7)!} = \dfrac{28!}{21!} = 22\cdot 23\cdot 24\cdot 25\cdot 26\cdot 27\cdot 28$

$a_{100} = \dfrac{\frac{100\cdot 101}{2}!}{\frac{99\cdot 100}{2}!} = \dfrac{(50\cdot 101)!}{(99\cdot 50)!} = \dfrac{5050!}{4950!} = 4951\cdot 4952\cdot ...\cdot 5050$