GENERALIZE-a-b-a-2-b-2-ab-a-3-b-3-a-b-c-a-2-b-2-c-2-ab-bc-ca-a-3-b-3-c-3-3abc-a-b-c-d-a-2-b-2-c-2-d-2-

Question Number 2240 by Rasheed Soomro last updated on 10/Nov/15

GENERALIZE :

(a + b) (a^{2} + b^{2} - a b) = a^{3} + b^{3}

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

= a^{3} + b^{3} + c^{3} - 3 a b c

(a + b + c + d) (a^{2} + b^{2} + c^{2} + d^{2} - \dots . .)

= a^{3} + b^{3} + c^{3} + d^{3} \dots \dots

- - - - - - - - - - - - - - - - - - - - - - - - -

(a_{1} + a_{2} + \dots + a_{n}) (a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2} - \dots . .)

= a_{1}^{3} + a_{2}^{3} + \dots + a_{n}^{3} \dots . .

Answered by Rasheed Soomro last updated on 15/Nov/15

(a + b + c + d) (a^{2} + b^{2} + c^{2} + d^{2} - a b - a c - a d - b c - b d - c d)

= a^{3} + b^{3} + c^{3} + d^{3} - 3 a b c - 3 a b d - 3 a c d - 3 b c d

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

(a + b + c + d) (a^{2} + b^{2} + c^{2} + d^{2} - a b - a c - a d - b c - b d - c d)

= a^{3} + b^{3} + c^{3} + d^{3} - 3 a b c - 3 a b d - 3 a c d - 3 b c d

\frac{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -}{}

(\underset{(1)}{a + b}) (\underset{(2)}{a^{2} + b^{2}} \underset{(3)}{- a b}) = \underset{(4)}{a^{3} + b^{3}} \underset{(5)}{- - - -}

(\underset{(1)}{a + b + c}) (\underset{(2)}{a^{2} + b^{2} + c^{2}} \underset{(3)}{- a b - a c - b c}) = \underset{(4)}{a^{3} + b^{3} + c^{3}} \underset{(5)}{- 3 a b c}

- - -

(\underset{(1)}{a + b + c + d}) (\underset{(2)}{a^{2} + b^{2} + c^{2} + d^{2}} \underset{(3)}{- a b - a c - a d - b c - b d - c d})

= \underset{(4)}{a^{3} + b^{3} + c^{3} + d^{3}} \underset{(5)}{- 3 a b c - 3 a b d - 3 a c d - 3 b c d}

O b s e r v e t h e PATTERN :

L H S

(1) : S u m o f a l l s y m b o l s .

(2) : S u m o f \underline{s q u a r e s o f a l l t h e s y m b o l s}

(3) : M i n u s s u m o f p r o d u c t s o f \underline{p o s s i b l e c o m b i n a t i o n s o f t w o}

R H S

(4) : S u m o f c u b e s o f a l l t h e s y m b o l s .

(5) : M i n u s t h r e e t i m e s t h e s u m o \underline{f p o s s i b l e c o m b i n a t i o n s o f t h r e e}

C o p p y i n g t h e PATTERN :

(\underset{(1)}{\overset{- - - - s u m - - - -}{a_{1} + a_{2} + \dots + a_{n}}}) (\underset{(2)}{\overset{- s u m o f s q u a r e s -}{a_{1}^{2} + a_{2}^{2} + \dots + a_{n}^{2}}} \underset{(3)}{\overset{p o s s i b l e c o m b i n a t i o n o f t w o s y m b o l s}{- a_{1} a_{2} - a_{1} a_{2} - \dots}} \dots \dots . .)

= \underset{(4)}{\overset{- - s u m o f c u b e s - -}{a_{1}^{3} + a_{2}^{3} + \dots + a_{n}^{3}}} \underset{(5)}{\overset{- - p o s s i b l e c o m b i n a t i o n o f t h r e e s y m b o l s - -}{- 3 a_{1} a_{2} a_{3} - 3 a_{1} a_{2} a_{4} - \dots .}} \dots \dots \dots \dots \dots