[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]
