CAT 2023 Slot 1 · Quant · Question 20

Write a=\sqrt{x}, b=\sqrt{y} and c=\sqrt{z} to keep the algebra short.

Being the arithmetic mean means twice the middle equals the sum of the outer two: \frac{2}{b+c}=\frac{1}{a+c}+\frac{1}{a+b}.

Combining the right side over a common denominator, \frac{1}{a+c}+\frac{1}{a+b}=\frac{\left( a+b \right)+\left( a+c \right)}{\left( a+c \right)\left( a+b \right)}=\frac{2a+b+c}{\left( a+c \right)\left( a+b \right)}.

Cross-multiplying, 2\left( a+c \right)\left( a+b \right)=\left( b+c \right)\left( 2a+b+c \right).

Expanding the left side, 2\left({a}^{2}+ab+ac+bc \right)=2{a}^{2}+2ab+2ac+2bc.

Expanding the right side, \left( b+c \right)\left( 2a+b+c \right)=2ab+{b}^{2}+bc+2ac+bc+{c}^{2}, that is 2ab+2ac+2bc+{b}^{2}+{c}^{2}.

Setting them equal, every mixed term cancels and 2{a}^{2}={b}^{2}+{c}^{2}.

Returning to the original letters, {a}^{2}=x, {b}^{2}=y and {c}^{2}=z, so 2x=y+z.

That says x is the arithmetic mean of y and z, so y, x, z are in arithmetic progression — the third option.

Shortcut. Rationalise each term instead: \frac{1}{\sqrt{y}+\sqrt{z}}=\frac{\sqrt{y}-\sqrt{z}}{y-z} and so on. Either route lands on 2x=y+z, and the answer must put x in the middle — which immediately rules out every option except the third.