How do you write a rule for the nth term of the geometric sequence and then find #a_5# given #a_4=189/1000, r=3/5#?

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Answer 1 · 2017-02-23T09:33:52

# a_5=\frac{7}{8} \times (\frac{3}{5})^4=\frac{567}{500}#

We know we can write every geometric sequence in the form of below:

#a_n = a_1 \times r^(n-1)#

and for now what we have to do is to figure out what is our first term ( #a_1#) and we can do it easily cause we have #a_4#:

#a_4=\frac{189}{1000}=a_1\times (\frac{3}{5})^3#

and we have to solve this equation for #a_1#

#a_1 = \frac{\frac{189}{1000}}{\frac{3^3}{5^3}} = \frac{189\times125}{1000\times27} = \frac{7}{8} #

Now we can rewrite our equation for any nth term:

#a_n = a_1 \times r^(n-1) = \frac{7}{8} \times (\frac{3}{5})^(n-1)#

and we can calculate #a_5# just by putting our #n=5# in the equation.

# a_5=\frac{7}{8} \times (\frac{3}{5})^4=\frac{567}{500}#