Question : $(113^2+ 115^2 + 117^2- 113×115 - 115×117 -117×113)$ is equal:
Option 1: 0
Option 2: 4
Option 3: 8
Option 4: 12
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Correct Answer: 12
Solution :
Use the formula:
$a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2}((a-b)^2 + (b-c)^2 + (c-a)^2)$
Now, $(113^2+ 115^2 + 117^2- 113×115 - 115×117 -117×113)$
$=\frac{1}{2}[(113-115)^2 + (115-117)^2 + (117-113)^2]$
$=\frac{1}{2}[(-2)^2 + (-2)^2 + (4)^2]$
$=\frac{1}{2}(4 + 4 + 16)$
$=\frac{1}{2}(24)$
$=12$
Hence, the correct answer is 12.
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