Explicație pas cu pas:
Observam ca suma ce trebuie calculata este suma unei progresii geometrice, unde primul termen (b1) este 1, iar ratia(q) este 1/3.
Formula sumei termenilor unei progresii geometrice:
[tex]S_n=b_1*\frac{q^n-1}{q-1}[/tex]
In cazul nostru n=11. Inlocuim in formula:
[tex]S_{11}=b_1*\frac{q^{11}-1}{q-1} \\\\S_{11}=1*\frac{(\frac{1}{3} )^{11}-1}{\frac{1}{3} -1} =-\frac{(\frac{1}{3}) ^{11}-1}{\frac{2}{3}}=\frac{1-(\frac{1}{3}) ^{11}}{\frac{2}{3}}=\frac{3*[1-(\frac{1}{3}) ^{11}]}{{2}}=\frac{3-(\frac{1}{3}) ^{10}]}{{2}}= > \\\\S_{11}=\frac{\frac{3^{10}-1}{3^{10}}}2= > \boxed{\boxed{S_{11}=\frac{3^{10}-1}{3^{10}*2} }}[/tex]
