Resposta :
Olá!!
[tex]a1=\frac {1} {4}\\ \\ q=\frac {1} {2} : \frac {1} {4}-->\frac {1.4} {2.1}=\frac {4} {2}-->(2)\\ \\ n=11\\ \\ Sn=?\\ \\ --------------------\\ \\ Sn=\frac {a1.(q^n-1)} {q-1}\\ \\ S7=\frac {1/4.(2^1^1-1)} {2-1}\\ \\ S7=\frac {1/4.(2048-1)} {1}\\ \\ S7=\frac {1} {4}.2047\\ \\ S7=\frac {2047} {4}[/tex]
Observe que:
[tex]\text{S}_{\text{n}}=\dfrac{\text{a}_1\cdot(\text{q}^{\text{n}}-1)}{\text{q}-1}[/tex]
Note que:
[tex]\text{q}=\dfrac{\text{a}_2}{\text{a}_1}=\dfrac{\frac{1}{2}}{\frac{1}{4}}=\dfrac{1}{2}\cdot\dfrac{4}{1}=2[/tex]
Logo:
[tex]\text{S}_{11}=\dfrac{\frac{1}{4}\cdot(2^{11}-1)}{2-1}=\dfrac{\frac{1}{4}\cdot2~047}{1}=\dfrac{2~047}{4}[/tex]
Portanto, a soma dos 11 primeiros termos da P.G. (1/4, 1/2, 1, 2, 4, ...) é [tex]\dfrac{2~047}{4}[/tex].