Determine if k − 1 k + 1 k ( k + 1 ) = k k + 1 holdsHow to prove if the following equality holds? k − 1 k + 1 k ( k + 1 ) = k k + 1 Maybe finding a common denominator would work, but I have no idea how to do it in this example.I see that it holds for k = 1 0 + 1 2 = 1 2 = 1 1 + 1 .It also holds for k = 2 1 2 + 1 6 = 3 6 + 1 6 = 4 6 = 2 3 = 2 2 + 1 It also works for k = 3 2 3 + 1 12 = 8 12 + 1 12 = 9 12 = 3 4 = 3 3 + 1 But I am not sure how to proceed if I am not working with numbers but with expressions.

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Determine if             k      −      1        k    +      1          k      (      k      +      1      )        =      k          k      +      1       holdsHow to prove if the following equality holds?            k      −      1        k    +      1          k      (      k      +      1      )        =      k          k      +      1      Maybe finding a common denominator would work, but I have no idea how to do it in this example.I see that it holds for   k  =  1  0  +      1    2    =      1    2    =      1          1      +      1        .It also holds for   k  =  2       1    2    +      1    6    =      3    6    +      1    6    =      4    6    =      2    3    =      2          2      +      1      It also works for   k  =  3      2    3    +      1    12    =      8    12    +      1    12    =      9    12    =      3    4    =      3          3      +      1      But I am not sure how to proceed if I am not working with numbers but with expressions.

postojahob · Accepted Answer

Notice that   k  (  k  +  1  )  is a common multiple of   k and   k  (  k  +  1  )Hence, for   k  ≠  0 and   k  ≠  −  1 we have            k      −      1        k    +      1          k      (      k      +      1      )        =                    k        2            −      1              k      (      k      +      1      )        +      1          k      (      k      +      1      )        =            k      2              k      (      k      +      1      )        =      k          k      +      1        .

Yesenia Obrien · Accepted Answer

You&#039;re having trouble with finding the common denominator. Let&#039;s compare the two denominators.            k      −      1        k    +      1          k      (      k      +      1      )       the difference between these two is one(on the right) is multiplied by   (  k  +  1  ) and the other(the left) is not. To make the denominators common we multiply the numerator and the denominator of the left fraction by             k      +      1              k      +      1       which if you think about it, is the same as multiplying it by one, which does not affect the value of the fraction. Doing this we get:            k      −      1        k    +      1          k      (      k      +      1      )        =                    k        2            −      1              k      (      k      +      1      )        +      1          k      (      k      +      1      )       (the 1s cancel when we add them)   =            k      ∗      k              k      (      k      +      1      )        =      k          k      +      1       (We cancel the ks on the top and bottom)It is important that this equality holds almost everywhere. We cannot say it is valid at   k  =  0  ,  −  1 this is because these values of k result in division by 0 which we know is not allowed.

Determine if k − 1 </mrow> k </mfrac>

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