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# The second term in a geometric sequence is 20. The fourth term in the same sequence is 11.25. What is the common ratio

let $a_n$ be the n'th term of the sequence. so $a_1$ is the first term, $a_2$ the second term and so on... In a geometric sequence with $a_1=c$, and common ratio r, the terms are as follows: $a_1=c$ $a_2=cr$ $a_3=c r^{2}$ $a_4=c r^{3}$ . . that is, each term is its previous term times the common ratio r. In our example $a_2=cr=20$ and $a_4=c r^{3}=11.25$ $\frac{a_2}{a_4}= \frac{cr}{c r^{3}}= \frac{1}{ r^{2}}= \frac{20}{11.25}= 1.778$ so $r^{2} =1/1.778=0.56$ $r= \sqrt{0.56}= 0.75$ Answer: r=0.75