Sequences & Series
AP and GP Conditions — Combined Constraint
nta_pyq_2023_apr
Grade 11
Question:
Let $0<z<y<x$ with $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ in AP and $x,\sqrt{2}y,z$ in GP. If $xy+yz+zx=\frac{3}{\sqrt{2}}xyz$, then $3(x+y+z)^2$ is equal to
Step-by-Step Solution
Key Concept: $\frac{2}{y}=\frac{1}{x}+\frac{1}{z}$ and $2y^2=xz$. From $\frac{xy+yz+zx}{xyz}=\frac{3}{\sqrt{2}}$: $\frac{1}{z}+\frac{1}{x}+\frac{1}{y}=\frac{3}{\sqrt{2}}\Rightarrow y=\sqrt{2}$.
$3(5\sqrt{2})^2=150$.
Correct Answer: 150