Question

Simplify the following cyclic expressions.

${\left(\sum _{a,b,c}a\right)}^2-\left(\sum _{a,b,c}{a}^2\right)$

• A. $(ab+bc+ca)$
• B. $2(ab+bc+ca)$
• C. $-2(ab+bc+ca)$
• D. None

Answer 1

The correct option is B $2(ab+bc+ca)$

Cyclic expressions refers to substitution of variables in cyclic manner.

We have, ${\left({\sum }_{a,b,c}a\right)}^2-\left({\sum }_{a,b,c}{a}^2\right)$

$=(a+b+c{)}^2-(a^2+b^2+c^2)$

$=(a^2+b^2+c^2+2ab+2bc+2ca)-(a^2+b^2+c^2)[\because (x+y+z{)}^2=x^2+y^2+z^2+2xy+2yz+2zx]$

$=2ab+2bc+2ca$

$=2(ab+bc+ca)$

Hence, $\text{B}$ is the correct option.