Question

Se da progresoa aritmetica a2+a9=10. S10=?

Answer 1

[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]$\text{Din datele pe care le avem nu le putem afla separat pe }a_1\text{ si }a_{10}$[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]

Answer 2

$a_1+a_{10} = 10\\ \bold{a_2+ a_9 = 10} \\ a_3+a_8 = 10 \\ a_4+a_7 = 10 \\ a_5+a_6 = 10 \\ --------(+) \\ a_1 + a_2 +...+a_{10} = 50\\ \\ \Rightarrow \boxed{S_{10} = 50}$