What is the common ratio of the sequence? -2. 6. -18. 54. …

For the geometric sequence, -frac{1}{54}, frac{1}{81}, -frac{2}{243},... (a) Find the first term and the common ratio for given sequence. (b) Find the general term, an for the given sequence. (c) Find a5, a10 and a20. (d) Find sum of first n terms, Sn for the given sequence. (d) Find S10 and S20.

Answer

-frac{1}{54}, frac{1}{81}, -frac{2}{243},... \First :term,a=-frac{1}{54} \Common :Ratio ,r=frac{frac{1}{81}}{-frac{1}{54}}=-frac{2}{3} \a_n=ar^{n-1}=-frac{1}{54}cdot(-frac{2}{3})^{n-1} \a_5=-frac{1}{54}cdot(-frac{2}{3})^{5-1}=-frac{frac{16}{81}}{54}=-frac{16}{4374} \a_{10}=-frac{1}{54}cdot(-frac{2}{3})^{10-1}=-frac{-frac{512}{19683}}{54}=frac{256}{531441} \a_{20}=-frac{1}{54}cdot(-frac{2}{3})^{20-1}=-frac{-frac{524288}{3^{19}}}{54}=frac{262144}{3^{19}cdot :27} \S_n=a(frac{1-r^n}{1-r}) \S_n=-frac{1}{54}cdot(frac{1-(-frac{2}{3})^n}{1-(-frac{2}{3})})=-frac{-left(-frac{2}{3}right)^n+1}{90} \S_{10}=-frac{-left(-frac{2}{3}right)^{10}+1}{90}=-frac{11605}{1062882}=-0.01092 \S_{20}=-frac{-left(-frac{2}{3}right)^{20}+1}{90}=-frac{697147165}{62762119218}=-0.01111

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