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+1 vote
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in Mathematics by (325 points)
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If,

a + b + c = 4

a2 + b2 + c2 = 10

a3 + b3 + c3 = 22

a4 + b4 + c4 = ? 

1 Answer

+1 vote
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Best answer

a + b + c = 4 ... (1)

⇒ a2 + b2 + c2 = 10 ... (2)

⇒ a3 + b3 + c3 = 22 ... (3)

Sq. both sides in (1) we get

⇒ (a + b + c)2 = (4)2

⇒ a2 + b2 + c2 + 2(ab + bc + ac) = 16

⇒ 10 + 2(ab + bc + ac) = 16 (From 2)

⇒ 2(ab + bc + ac) = 6

⇒ (ab + bc + ac) = 3 ... (4)

Consider the identity, 

a3 + b3 + c3 - 3abc = (a + b + c) (a2 + b2 + c2 - {ab + bc + ac}) 

⇒ 22 - 3abc = 4(10 - 3) ... (From 1,2,3,4)

⇒ 22 - 3abc = 28

⇒ -3abc = 6

⇒ abc = -2 ... (5) 

Sq. both sides in (4), we get

⇒ (ab + bc + ac)2 = (3)2

⇒ (ab)2 + (bc)2 + (ac)2 + 2b2ac + 2c2ab + 2a2bc = 9

⇒ (ab)2 + (bc)2 + (ac)+ 2abc(a + b + c) = 9

⇒ (ab)2 + (bc)2 + (ac)2 + 2(-2)(4) = 9 ... (From 5, 1)

⇒ (ab)2 + (bc)2 + (ac)- 16 = 9

⇒ (ab)2 + (bc)2 + (ac)2 = 25 ... (6)

Sq. both sides in (2) we get

⇒ (a2 + b2 + c2)2 = (10)2

⇒ a4 + b4 + c4 + 2{(ab)2 + (bc)2 + (ac)2} = 100

⇒ a4 + b4 + c4 + 2(25) = 100 ... (From 6)

⇒ a4 + b4 + c+ 50 = 100

∴ a4 + b4 + c= 50

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