Question

Se da sirul: 2;7;16;29;46;67. Sa se afle al 100-lea termen

Answer 1

$2;7;16;29;46;67;....\\ \\ a_1 = 2\\a_2 = a_1+5\\ a_3 = a_2+9\\ a_4 = a_3+13 \\ a_5 = a_4+17\\ a_6 = a_5+21 \\ ...\\ a_n = a_{n-1}+(4n-3)\\ \\ \\ a_1+a_2+...+a_n = a_1+a_2+...+a_{n-1}+2+5+9+13+...+(4n-3) \\ \\ a_n = 2+5+9+13+...+(4n-3) = \\ \\ = 1+\Big(1+5+9+13+...+(4n-3)\Big) = \\ \\ = 1+\Big(1+(2+3)+(4+5)+(6+7)+...+(2n-2+2n-1)\Big) = \\ \\ = 1+\dfrac{(2n-1)(2n-1+1)}{2} = 1+\dfrac{(2n-1)\cdot 2n}{2} = 1+(2n-1)n \\ \\ \Rightarrow a_n = 1+n(2n-1)$

$\Rightarrow a_{100} = 1+100\cdot (2\cdot 100-1) = 1+100\cdot 199 \\ \\ \Rightarrow \boxed{a_{100} = 19901}$