Question

The fourth, seventh, and tenth terms of a G.P. are \( p, q, r \) respectively, then:

• A. \( p^2 = q^2 + r^2 \)
• B. \( q^2 = pr \)
• C. \( p^2 = qr \)
• D. \( pqr + pq + 1 = 0 \)

Hint:

In a geometric progression, the relationships between terms can be simplified using the common ratio and the formula for the general term.

Answer 1

The Correct Option is B

Step 1: The general term of a G.P. is given by: \[ T_n = ar^{n-1}, \] where \( a \) is the first term and \( r \) is the common ratio. The fourth, seventh, and tenth terms are: \[ p = ar^3, \quad q = ar^6, \quad r = ar^9. \] Step 2: To find the relationship between \( p, q, r \), we divide \( q^2 \) by \( pr \): \[ \frac{q^2}{pr} = \frac{(ar^6)^2}{(ar^3)(ar^9)} = \frac{a^2r^{12}}{a^2r^{12}} = 1. \] Thus, \( q^2 = pr \).