If a,b,c are in gp. prove that a2+b2,ab+bc(b2+c2 are in gp
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Step-by-step explanation:
If a,b,c are in gp. prove that a2+b2, ab+bc , b2+c2 are in gp
a , b , c are in GP
=> b² = ac
a² + b² , ab + bc , b² + c² will be in GP
iff
( ab + bc )² = (a² + b²)( b² + c² )
a²b² + b²c² + 2ab²c = a²b² + a²c² + b⁴ + b²c²
2ab²c = a²c² + b⁴
if we put b² = ac
=> 2ac*ac = a²c² + (ac)²
=> 2a²c² = 2a²c²
LHS = RHS
Hence a² + b² , ab + bc , b² + c² will be in GP if a , b . c are in GP
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