Question

Cine imi poate face si mie aceste exercitii?

Answer 1

   
[tex]\displaystyle\\ 3)\\ S = 5+8+11+14+\cdots + 101\\\\ \text{Acesta este un sir Gauss}.\\\\ \text{Calculam numarul de termeni:}\\\\ n = \frac{101-5}{3}+1 = \frac{96}{3}+1 =32+1 = 33~\text{ de termeni}\\\\ S = 5+8+11+14+\cdots + 101 =\\\\ = \frac{n(101+5)}{2}= \frac{33\times106}{2}=33\times53 = \boxed{\bf 1749} \\\\ 4)\\ T=1+3+3^2+3^3+\cdots + 3^{20} = \frac{3^{20+1}-1}{3-1}= \boxed{\bf \frac{3^{21}-1}{2}}[/tex]


[tex]\displaystyle\\ 5)\\ S= 1+3+5+\cdots + (2n+1) \\\\ \text{Acesta este un sir Gauss}.\\\\ \text{Calculam numarul de termeni:}\\\\ n = \frac{(2n+1)-1}{2}+1 = \frac{2n+1-1}{2}+1 = \frac{2n}{2}+1 =n+1 ~~\text{ termeni}\\\\ S= 1+3+5+\cdots + (2n+1)=\\\\ = \frac{(n+1)[(2n+1)+1]}{2}= \frac{(n+1)(2n+1+1)}{2}=\\\\ =\frac{(n+1)(2n+2)}{2} = \frac{(n+1)(n+1)\cdot 2}{2}= (n+1)(n+1)=\boxed{\bf(n+1)^2}\\\\\\ 6)\\ S = 1+a+a^1+a^2+\cdots+a^n= \boxed{\bf\frac{a^{n+1}-1}{a-1}} [/tex]