# Detemine whether the given sequence geometric or neither if the sequence is srithmetic, find the common difference, if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum text{of} left{left(frac{5}{8}right)^{n}right}

Question
Polynomial arithmetic
Detemine whether the given sequence geometric or neither if the sequence is srithmetic, find the common difference, if it is geometric, find the common ratio.
If the sequence is arithmetic or geometric, find the $$\sum\ \text{of}\ \left\{\left(\frac{5}{8}\right)^{n}\right\}$$

2020-12-01
Step 1
The given sequence is exponential, it has n in the exponent. So this is a geometric sequence.
Common ratio of the sequence is:
$$\frac{second\ term}{first\ term}=\frac{\left(\frac{5}{8}\right)^{2}}{\left(\frac{5}{8}\right)^{1}}=\frac{5}{8}$$
Step 2
We will use the $$\sum$$ formula of the geometric series.
$$S_{n}=\frac{a^{1}(1\ -\ r^{n})}{1\ -\ r}$$
$$S_{50}=\frac{\frac{5}{8}\left(1\ -\ \left(\frac{5}{8}\right)^{50}\right)}{1\ -\ \frac{5}{8}}$$
$$S_{50}=\ \frac{\frac{5}{8}\ (0.999999999938)}{\frac{3}{8}},\ S_{50}=\ \frac{5(0.999999999938)}{3}$$
$$S_{50}=1.667$$
Answer: Geometric, $$\text{common ratio}=\frac{5}{8},\ \sum=1.667$$

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