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

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

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