Найдите сумму первых пяти членов геометрической прогрессии (bn), в которой: b4=1/16, b5=1/64.

\[S_{5} = \frac{b_{1}\left( q^{5} - 1 \right)}{q - 1}\]

\[b_{4} = \frac{1}{16};\ \ b_{5} = \frac{1}{64}:\]

\[q = \frac{b_{5}}{b_{4}} = \frac{1}{64}\ :\frac{1}{16} = \frac{1}{64} \cdot 16 = \frac{1}{4}.\]

\[b_{4} = b_{1} \cdot q^{3}\]

\[b_{1} = \frac{b_{4}}{q^{3}} = \frac{1}{16}\ :\left( \frac{1}{4} \right)^{3} = \frac{1}{16} \cdot 64 =\]

\[= 4.\]

\[S_{5} = \frac{4 \cdot \left( \left( \frac{1}{4} \right)^{5} - 1 \right)}{\frac{1}{4} - 1} =\]

\[= \frac{4 \cdot \left( \frac{1}{1024} - 1 \right)}{- \frac{3}{4}} = \frac{4 \cdot \frac{1023}{1024}}{\frac{3}{4}} =\]

\[= \frac{4 \cdot 1023}{3 \cdot 256} =\]

\[= \frac{341}{64} = 5\frac{21}{64}.\]