Question

Given that, вв
( a, b, c ) are in ( A P )
( Leftrightarrow(b-a)=(-b) )
1
i) ( a^{2}(b+c), b^{2}(c+a), c^{2}(a+b) )
dets subtract fist and middle term and eastand middle tom
( [a] quad b^{2}(c+a)-a^{2}(b+c) )
[
begin{array}{l}
=b^{2} c+a b^{2}-a^{2} b^{2}-a^{2} c
=a b(b-a)+cleft(b^{2}-a^{2}right)
=a b(b-a)+c(b-a)(b+a)
=(b-a)[a b+a c+b c]-
end{array}
]
( [b] quad c^{2}(a+b)-b^{2}(c+a) )
[
begin{array}{l}
=c^{2} a+c^{2} b-b^{2} c-b^{2} a
=left(a-b^{2}+b^{2}right)+b c(c-b)
=a(c-b)(c+b)+b c(c-b)
=(c-b)[a c+a b+b c]-
end{array}
]
7 we equate (a) and(b) we get eq ( ^{n}(1) ) ( (b-a)=(c-b)- ) we got eq ( ^{n}(1) )
thence, the given terms are in ( A P ). to

# If a, b, c are in A.P., then show that: () a (b + c), b2 (c + a), c2 (a + b) are also in A.P.

Solution