[tex]S_{10}=?\\\\
\text{Formula pentru suma primilor n termeni a unei progresii aritmetice:}\\\\
S_n=\frac{(a_1+a_n)\cdot n}{2}\\
n=10\rightarrow S_{10}=\frac{(a_1+a_{10})\cdot10}{2}=\boxed{5(a_1+a_{10})}\\\\
\text{Pentru a putea aplica formula trebuie sa stim } a_1 \text{ si } a_{10}.\\
[/tex][tex]\text{Din datele pe care le avem nu le putem afla separat pe }a_1\text{ si }a_{10}\\
[/tex][tex] \text{dar noi avem nevoie doar de } (a_1 + a_{10}).
[/tex][tex]\text{Formula termenului general a unei progresii aritmetice:}\\
a_n=a_1+(n-1)r[/tex][tex]a_2+a_9=10
\\
a_2+a_9=(a_1+1\cdot r)+(a_1+8\cdot r)=2a_1+9r\\
\boxed{2a_1+9r=10}[/tex][tex]a_1+a_{10}=a_1+(a_1+9\cdot r)=2a_1+9r=10\\
\boxed{a_1+a_{10}=10}[/tex][tex]\text{Acum ca am aflat } (a_1+a_{10})\text{ inlocuim in formula sumei}:\\\\
S_{10}=5(a_1+a_{10})=5\cdot 10=50\\
\boxed{S_{10}=50}[/tex]
