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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-

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Question Number 2240 by Rasheed Soomro last updated on 10/Nov/15
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 −.....) =a^3 +b^3 +c^3 +d^3 ...... −−−−−−−−−−−−−−−−−−−−−−−−− (a_1 +a_2 +...+a_n )(a_1 ^( 2) +a_2 ^( 2) +...+a_n ^( 2) −.....) =a_1 ^( 3) +a_2 ^( 3) +...+a_n ^( 3) .....
$$\mathcal{GENERALIZE}: \\ $$$$\left({a}+{b}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{ab}\right)={a}^{\mathrm{3}} +{b}^{\mathrm{3}} \\ $$$$\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} −\mathrm{3}{abc} \\ $$$$\left({a}+{b}+{c}+{d}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} −…..\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{d}^{\mathrm{3}} \:…… \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$$\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +…+{a}_{{n}} \right)\left({a}_{\mathrm{1}} ^{\:\mathrm{2}} +{a}_{\mathrm{2}} ^{\:\mathrm{2}} +…+{a}_{{n}} ^{\:\mathrm{2}} −…..\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}_{\mathrm{1}} ^{\:\mathrm{3}} +{a}_{\mathrm{2}} ^{\:\mathrm{3}} +…+{a}_{{n}} ^{\:\mathrm{3}} \:….. \\ $$
Answered by Rasheed Soomro last updated on 15/Nov/15
(a+b+c+d)(a^2 +b^2 +c^2 +d^2 −ab−ac−ad−bc−bd−cd) =a^3 +b^3 +c^3 +d^3 −3abc−3abd−3acd−3bcd −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− (a+b+c+d)(a^2 +b^2 +c^2 +d^2 −ab−ac−ad−bc−bd−cd) =a^3 +b^3 +c^3 +d^3 −3abc−3abd−3acd−3bcd ((−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−)/) (a+b_((1)) )(a^2 +b^2 _((2)) −ab_((3)) )=a^3 +b^3 _((4)) −−−−_((5)) (a+b+c_((1)) )(a^2 +b^2 +c^2 _((2)) −ab−ac−bc_((3)) )=a^3 +b^3 +c^3 _((4)) −3abc_((5)) −−− (a+b+c+d_((1)) )(a^2 +b^2 +c^2 +d^2 _((2)) −ab−ac−ad−bc−bd−cd_((3)) ) =a^3 +b^3 +c^3 +d^3 _((4)) −3abc−3abd−3acd−3bcd_((5)) Observe the PATTERN: LHS (1):Sum of all symbols. (2): Sum of squares of all the symbols_(−) (3):Minus sum of products of possible combinations of two_(−) RHS (4):Sum of cubes of all the symbols. (5):Minus three times the sum of possible combinations of three _(−) Coppying the PATTERN: (a_1 +a_2 +...+a_n _((1)) ^(−−−−sum−−−−) )(a_1 ^( 2) +a_2 ^( 2) +...+a_n ^( 2) _((2)) ^(−sum of squares−) −a_1 a_2 −a_1 a_2 −..._((3)) ........^(possible combination of two symbols) ) =a_1 ^( 3) +a_2 ^( 3) +...+a_n ^( 3) _((4)) ^(−−sum of cubes−−) −3a_1 a_2 a_3 −3a_1 a_2 a_4 −...._((5)) ...............^(−−possible combination of three symbols−−)
$$\left({a}+{b}+{c}+{d}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} −{ab}−{ac}−{ad}−{bc}−{bd}−{cd}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{d}^{\mathrm{3}} −\mathrm{3}{abc}−\mathrm{3}{abd}−\mathrm{3}{acd}−\mathrm{3}{bcd} \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− \\ $$$$\left({a}+{b}+{c}+{d}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} −{ab}−{ac}−{ad}−{bc}−{bd}−{cd}\right) \\ $$$$\:\:\:={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{d}^{\mathrm{3}} −\mathrm{3}{abc}−\mathrm{3}{abd}−\mathrm{3}{acd}−\mathrm{3}{bcd} \\ $$$$\frac{−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−}{} \\ $$$$\left(\underset{\left(\mathrm{1}\right)} {{a}+{b}}\right)\left(\underset{\left(\mathrm{2}\right)} {{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\underset{\left(\mathrm{3}\right)} {−{ab}}\right)=\underset{\left(\mathrm{4}\right)} {{a}^{\mathrm{3}} +{b}^{\mathrm{3}} }\underset{\left(\mathrm{5}\right)} {−−−−} \\ $$$$\left(\underset{\left(\mathrm{1}\right)} {{a}+{b}+{c}}\right)\left(\underset{\left(\mathrm{2}\right)} {{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} }\underset{\left(\mathrm{3}\right)} {−{ab}−{ac}−{bc}}\right)=\underset{\left(\mathrm{4}\right)} {{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} }\underset{\left(\mathrm{5}\right)} {−\mathrm{3}{abc}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−− \\ $$$$\left(\underset{\left(\mathrm{1}\right)} {{a}+{b}+{c}+{d}}\right)\left(\underset{\left(\mathrm{2}\right)} {{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} +{d}^{\mathrm{2}} }\underset{\left(\mathrm{3}\right)} {−{ab}−{ac}−{ad}−{bc}−{bd}−{cd}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:=\underset{\left(\mathrm{4}\right)} {{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} +{d}^{\mathrm{3}} }\underset{\left(\mathrm{5}\right)} {−\mathrm{3}{abc}−\mathrm{3}{abd}−\mathrm{3}{acd}−\mathrm{3}{bcd}} \\ $$$$ \\ $$$$\mathcal{O}{bserve}\:{the}\:\mathcal{PATTERN}: \\ $$$${LHS} \\ $$$$\left(\mathrm{1}\right):{Sum}\:{of}\:{all}\:{symbols}. \\ $$$$\left(\mathrm{2}\right):\:{Sum}\:{of}\:\underset{−} {{squares}\:{of}\:{all}\:{the}\:{symbols}} \\ $$$$\left(\mathrm{3}\right):{Minus}\:{sum}\:{of}\:{products}\:\:{of}\:\:\:\underset{−} {\:\:{possible}\:{combinations}\:{of}\:{two}} \\ $$$${RHS} \\ $$$$\left(\mathrm{4}\right):{Sum}\:{of}\:{cubes}\:{of}\:{all}\:{the}\:{symbols}. \\ $$$$\left(\mathrm{5}\right):{Minus}\:{three}\:{times}\:{the}\:{sum}\:{o}\underset{−} {{f}\:\:{possible}\:{combinations}\:{of}\:{three}\:\:\:} \\ $$$$\mathcal{C}{oppying}\:{the}\:\mathcal{PATTERN}: \\ $$$$\left(\underset{\left(\mathrm{1}\right)} {\overset{−−−−{sum}−−−−} {{a}_{\mathrm{1}} +{a}_{\mathrm{2}} +…+{a}_{{n}} }}\right)\left(\underset{\left(\mathrm{2}\right)} {\overset{−{sum}\:{of}\:{squares}−} {{a}_{\mathrm{1}} ^{\:\mathrm{2}} +{a}_{\mathrm{2}} ^{\:\mathrm{2}} +…+{a}_{{n}} ^{\:\mathrm{2}} }}\underset{\left(\mathrm{3}\right)} {\overset{{possible}\:{combination}\:{of}\:{two}\:{symbols}} {−{a}_{\mathrm{1}} {a}_{\mathrm{2}} −{a}_{\mathrm{1}} {a}_{\mathrm{2}} −…}}……..\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\underset{\left(\mathrm{4}\right)} {\overset{−−{sum}\:{of}\:{cubes}−−} {{a}_{\mathrm{1}} ^{\:\mathrm{3}} +{a}_{\mathrm{2}} ^{\:\mathrm{3}} +…+{a}_{{n}} ^{\:\mathrm{3}} }}\underset{\left(\mathrm{5}\right)} {\overset{−−{possible}\:{combination}\:{of}\:{three}\:{symbols}−−} {−\mathrm{3}{a}_{\mathrm{1}} {a}_{\mathrm{2}} {a}_{\mathrm{3}} −\mathrm{3}{a}_{\mathrm{1}} {a}_{\mathrm{2}} {a}_{\mathrm{4}} −….}}…………… \\ $$

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