If a1, a2, a3 (with .a1>0) are in G.P. with common ratio r, then the value of r for which the in

step 1 Since A1, A2, and A3 are three consecutive terms in the geometric progression, so we assume the

If a1, a2, a3 (with .a1>0) are in G.P. with common ratio r,...

If $a_{1}, a_{2}, a_{3}$ (with $\left.a_{1}>0\right)$ are in G.P. with common ratio $r$, then the value of $r$ for which the inequality $9 a_{1}+5 a_{3}$ $>14 a_{2}$ holds, cannot be in the interval (A) $\left[1, \frac{9}{2}\right]$ (B) $(-\infty, 0)$ (C) $\left[\frac{5}{9}, 1\right]$ (D) $\left[1, \frac{9}{5}\right]$

If $a_{1}, a_{2}, a_{3}$ (with $\left.a_{1}>0\right)$ are in G.P. with common ratio $r$, then the value of $r$ for which the inequality $9 a_{1}+5 a_{3}$ $>14 a_{2}$ holds, cannot be in the interval
(A) $\left[1, \frac{9}{2}\right]$
(B) $(-\infty, 0)$
(C) $\left[\frac{5}{9}, 1\right]$
(D) $\left[1, \frac{9}{5}\right]$

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Key Concepts

Inequality Solving

Inequality solving involves determining the range of values that satisfy a given inequality. This usually requires algebraic manipulation and consideration of the sign of expressions, especially when those expressions involve quadratic terms obtained from the structure of the sequence.

Quadratic Equations and Inequalities

Many problems involving geometric progressions lead to quadratic equations or inequalities when expressing conditions between terms. Understanding how to form, factor, and solve quadratic expressions is important for analyzing when certain conditions, such as inequalities, hold.

Geometric Progressions

A geometric progression is a sequence in which each term is obtained by multiplying the preceding term by a constant factor called the common ratio. This concept is foundational in understanding patterns and behaviors in sequences and series.

Common Ratio

The common ratio is the constant factor by which successive terms in a geometric progression are multiplied. Recognizing and manipulating the common ratio is crucial when analyzing the properties of the sequence.

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