Question
If x = 997, y = 998 and z = 999 then the value of x2 + y2 + z2 - xy - yz - zx is?
Answer: Option D
$$\eqalign{
& {x^2} + {y^2} + {z^2} - xy - yz - zx \cr
& = \frac{1}{2}\left[ {{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}} \right] \cr} $$
$$ = \frac{1}{2}$$ $$\left[ {{{\left( {997 - 998} \right)}^2} + {{\left( {998 - 999} \right)}^2} + {{\left( {999 - 997} \right)}^2}} \right]$$
$$\eqalign{
& = \frac{1}{2}\left( {1 + 1 + 4} \right) \cr
& = 3 \cr} $$
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$$\eqalign{
& {x^2} + {y^2} + {z^2} - xy - yz - zx \cr
& = \frac{1}{2}\left[ {{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}} \right] \cr} $$
$$ = \frac{1}{2}$$ $$\left[ {{{\left( {997 - 998} \right)}^2} + {{\left( {998 - 999} \right)}^2} + {{\left( {999 - 997} \right)}^2}} \right]$$
$$\eqalign{
& = \frac{1}{2}\left( {1 + 1 + 4} \right) \cr
& = 3 \cr} $$
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