Question 17

If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in G.P.

Answer

Let a be the first term and r be the common ratio of the G.P.

According to the given condition,

a₄ = a r³ = x … (1)

a₁₀ = a r⁹ = y … (2)

a₁₆ = a r¹⁵ = z … (3)

Dividing (2) by (1), we obtain

$y over x space equals space fraction numerator a r to the power of 9 over denominator a r cubed end fraction space rightwards double arrow y over x space equals space r to the power of 6$

Dividing (3) by (2), we obtain

$z over y space equals space fraction numerator a r to the power of 15 over denominator a r to the power of 9 end fraction rightwards double arrow z over y equals r to the power of 6$