Question

Simplify the following cyclic expressions.
(i) ${\left({\sum }_{a,b,c}a\right)}^4-{\sum }_{a,b,c}(a+b{)}^4+{\sum }_{a,b,c}(a{)}^4$
(ii) ${\sum }_{a,b,c}(a+1)(a+2)(b-c)$

Answer 1

(i) $(a+b+c{)}^4-(a+b{)}^4-(b+c{)}^4-(c+a{)}^4+(a^4+b^4+c^4)$
$(a+b+c{)}^4-(a+b{)}^4-(b+c{)}^4-(c+a{)}^4+a^4+b^4+c^4$

(ii) $(a+1)(a+2)(b-c)+(b+1)(b+2)(c-a)+(c+1)(c+2)(a-b)$