Question

Only Don like me can solve it ⬆⬆

If a, b, c are in HP then proove that , ( 1/b + 1/c - 1/a ) , ( 1/a + 1/c - 1/b ) = 2/bc - 1/b² = 3/b² - 2/ab

Answer 1

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By the way i am not don . well thanks for asking question . I will try to give my best !!!

Solution :-

if 1 / a , 1/b , 1/c are in H . P

then , 2/b = 1/a + 1/c

1/a = 2/b - 1/c --------1)

but , ( 1/b + 1/c - 1/a) . ( 1/a + 1/c - 1/b) = ?

From ( 1)

= ( 1 / b + 1/c - 2/b + 1/c ) . ( 2/b - 1/c + 1/c - 1/b )

= ( 2/c - 1/b ) (1/b ) = 2/bc - 1/b² 1st prooved ♻

again,

= 2/b = ( 1/a + 1/c )

= 2/b - 1/a = 1/c ----- 2nd

(since from 2nd )

= ( 1/b + 2 / b - 1/a - 1/a ) (/ ( 1/a + 2/b - 1/a - 1/b )

= (3/b - 2/a ). (1/b ) = 3/b² - 2/ab

here prooved ♻

( 1/b + 1/c - 1/a) . ( 1/a + 1/c - 1/b ) = 2bc - 1/b² = 3/b² - 2/ab

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