Sequences & Series
Sequence and Series
star_batch_jee_advanced_2025
Grade 11
Question:
Let $x, y, z$ are positive real numbers satisfy $2x - 2y + \frac{1}{z} = \frac{1}{2018}, 2y - 2z + \frac{1}{x} = \frac{1}{2018}, 2z - 2x + \frac{1}{y} = \frac{1}{2018}$ then $x + y - z$ is equal to ____.
Step-by-Step Solution
Key Concept: By adding the three derived equations and using the constraint on the sum of reciprocals, symmetry forces $x = y = z = 2018$.
Given $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{3}{2018}$, three equations are derived: $2xz - 2yz + 1 = \frac{z}{2018}$, $2yz - 2zx + 1 = \frac{x}{2018}$, and $2yz - 2xy + 1 = \frac{y}{2018}$. Adding all three and simplifying with $x + y + z = 2018 \times 3$ yields $\frac{3}{\frac{1}{x} + \frac{1}{y} + \frac{1}{z}} = \frac{x+y+z}{3}$, which gives $x = y = z = 2018$.
Correct Answer: 2018