if a,b,c,d are an arithmetic progression (in that order), prove that (1)/(sqrt a+sqrt b)+(1)/(sqrt b +sqrt c) +(1)/(sqrt c + \sqrt d) = (3)/(sqrt a + sqrt d)

Awainaideannagi 2022-07-21 Answered
proving that 1 a + b + 1 b + c + 1 c + d = 3 a + d for any A.P.
if a , b , c , d are an arithmetic progression (in that order), prove that
I made n the common difference of a , b , c , d; so
a = a
b = a + n
c = a + 2 n
d = a + 3 n
I tried to replace the terms with those, anyways i squared both equalities but i didn 't get nothing since i'm pretty bad with square roots. I'm looking for some hints or properties that can be useful. Thanks
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Answers (2)

Kaiden Weeks
Answered 2022-07-22 Author has 14 answers
HINT
Multiply and divide by conjugate of each denominator, then you'll get a ( n ) in each denominator.
Then :
a b   + b c   + c d n
= a d n = a d n ( a + d ) = 3 a + d

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Raynor2i
Answered 2022-07-23 Author has 6 answers
1 a + b + 1 b + c + 1 c + d = 3 a + d
Make into A.P.
S = 1 a + a + x + 1 a + x + a + 2 x + 1 a + 2 x + a + 3 x 3 a + a + 3 x
Rationalize
1 a + a + x a a + x a a + x = a a + x x
1 a + x + a + 2 x a + x a + 2 x a + x a + 2 x = a + x a + 2 x x
1 a + 2 x + a + 3 x a + 2 x a + 3 x a + 2 x a + 3 x = a + 2 x a + 3 x x
3 a + a + 3 x a a + 3 x a a + 3 x = 3 a 3 a + 3 x 3 x = a a + 3 x x
Combine
S = ( 1 / x ) ( ( a a + x ) + ( a + x a + 2 x ) + ( a + 2 x a + 3 x ) ( a a + 3 x ) ) = 0 because everything cancels out!!!

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