Question

What is the common ratio of the geometric sequence whose second and fourth terms are 6 and 54, respectively?

Answer 1

A₂ = 6      a₄ = 54

$a_{n} = q^{n-1} * a_{1}$ 

$\left \{ {{ a_{2} = q^{2-1} * a_{1} } \atop { a_{4} = q^{4-1}* a_{1} }} \right. \\ \\ \left \{ {{6 = q * a_{1} } \atop {54 = q^{3} * a_{1} }} \right. \\ \\ \left \{ {{ a_{1} = \frac{6}{q} } \atop {54 = q^{3} * \frac{6}{q} }} \right. \\ \\ \left \{ {{ a_{1} = \frac{6}{q} } \atop {54 = q^{2} * 6 }} \right.$ 

$\left \{ {{ a_{1} = \frac{6}{q} } \atop { q^{2} =9}} \right. \\ \\ \left \{ {{ a_{1} = \frac{6}{q} } \atop {q= \sqrt{9} }} \right.$ 
q = 3     q = -3

a₁ = 6/3 = 2   a₁ = 6/-3 = -2

Answer 2

Hi there! T4=T2×r²,6r²=54. Therefore, the answer would be 3.