The sum of the infinite geometric series ( x − 1 ) + ( x − 1 ) 2 + ( x − 1 ) 3 . . . i s . . . 1 / 3 . Determine the value of x.

Question

The sum of the infinite geometric series   (  x  −  1  )  +  (  x  −  1      )    2    +  (  x  −  1      )    3    .  .  .  i  s  .  .  .  1      /    3 . Determine the value of x.

Grace Moses · Accepted Answer

The sum of the infinite geometric series   (  x  −  1  )  +  (  x  −  1      )    2    +  (  x  −  1      )    3    .  .  .  i  s  .  .  .      1    3  for solving the problem let us assume that thesum is Sso,   S  =  (  x  −  1  )  +  (  x  −  1      )    2    +  (  x  −  1      )    3    +  (  x  −  1      )    4    +    =  (  x  −  1  )  +  (  x  −  1  )  ⋆  [  (  x  −  1  )  +  (  x  −  1      )    2    +  (  x  −  1      )    3    +  ]    =  (  x  −  1  )  +  (  x  −  1  )  ⋆  S aswe know that,  S  =  (  x  −  1  )  +  (  x  −  1      )    2    +  (  x  −  1      )    3    +  (  x  −  1      )    4    +    =  (  x  −  1  )  ⋆  (  1  +  S  )    S  =  (  x  −  1  )  ⋆  (  1  +  S  )  (  1  )in our problem, S =       1    3  ,so, we put S =       1    3   in (1) to get,      1    3    =  (  x  −  1  )  ⋆  (  1  +      1    3    )  ⇒      1    3    =  (  x  −  1  )  ⋆      4    3    ⇒ 1 = 4x - 4  ⇒ 4x = 1+4 = 5  ⇒ x =       5    4  Answer:- x =       5    4

The sum of the infinite geometric series(x-1)+(x-1)^2+(x-1)^3....is 1/3. Determine the value ofx.

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