Question

The 5th, 8th and 11th terms of a G.P. are p, q and s, respectively. Show that q2 = ps.

Answer 1

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

According to the given condition,

a5 = a r 5–1 = a r4 = p … (1)

a8 = a r 8–1 = a r 7 = q … (2)

a11 = a r 11†“1 = a r 10 = s … (3)

Dividing equation (2) by (1), we obtain

$\frac{ar7}{ar4}=\frac{q}{p}$

r3 = $\frac{q}{p}$ ... (4)

Dividing equation (3) by (2), we obtain

$\frac{ar10}{ar7}=\frac{s}{q}$

⇒ r3 =$\frac{s}{q}$ ... (5)

Equating the values of r 3 obtained in (4) and (5), we obtain

$\frac{q}{p}=\frac{s}{q}$

⇒ q2 = ps

Thus, the given result is proved.