Sequences & Series
Sequences And Series
nta_abhyas_2025
Grade 11
Question:
Let $\alpha$ and $\beta$ be two numbers where $\alpha < \beta$. The geometric mean of these numbers exceeds the smaller number $\alpha$ by 12 and the arithmetic mean of the same numbers is smaller by 24 than the larger number $\beta$, then the value of $|\beta - \alpha|$ is
Step-by-Step Solution
Key Concept: Set up equations using the definitions of geometric mean and arithmetic mean, then solve the resulting system.
From the geometric mean condition: $\sqrt{\alpha\beta} = \alpha + 12$. From the arithmetic mean condition: $\frac{\alpha + \beta}{2} = \beta - 24$. The second equation gives $\alpha + \beta = 2\beta - 48$, so $\alpha = \beta - 48$. Substituting into the first equation and solving yields $\alpha = 6, \beta = 54$. Therefore $|\beta - \alpha| = |54 - 6| = 48$.
Correct Answer: 48