In a GP, the 3rd, 5th and 7th terms are \(x\), \(x^2+2\) and \(x^3+10\) respectively, where \(x>1\). What is the 8th term?
\(18\sqrt{3}\)
Since the 3rd, 5th and 7th terms are equally spaced in a GP, \((x^2+2)^2 = x(x^3+10)\), which simplifies to \(2x^2-5x+2=0\), giving \(x=2\) (since \(x>1\)). Then the 3rd term is 2 and the common ratio square is \(r^2 = 6/2 = 3\) (using the 5th term \(=6\)), so \(r=\sqrt{3}\) and the first term is \(2/3\). The 8th term is \(\dfrac{2}{3}\cdot(\sqrt{3})^7 = 18\sqrt{3}\).
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