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 thatI made n the common difference of a , b , c , d; so a = a b = a + n c = a + 2 n d = a + 3 nI 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

Question

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 thatI made   n the common difference of   a  ,  b  ,  c  ,  d; so  a  =  a  b  =  a  +  n  c  =  a  +  2  n  d  =  a  +  3  nI tried to replace the terms with those, anyways i squared both equalities but i didn &#039;t get nothing since i&#039;m pretty bad with square roots. I&#039;m looking for some hints or properties that can be useful. Thanks

Kaiden Weeks · Accepted Answer

HINTMultiply and divide by conjugate of each denominator, then you&#039;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

Raynor2i · Accepted Answer

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!!!

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)

Answered question

Answer & Explanation

New Questions in Pre-Algebra