Simplifying equations

[jones123]

Hi,

this may seem silly but I can't figure out why these equations are equal...

1) Why is

(1)/(k+1) + (1)/(k-1) = (2k)/(k²-1) ?

2) Why is

[(-1)^k / (k+1)] + [(-1)^k / (k-1)] = ((-1)^k) + 1 ?

Thanks !

 

[stapel]

1) Why is

(1)/(k+1) + (1)/(k-1) = (2k)/(k²-1) ?

2) Why is

[(-1)^k / (k+1)] + [(-1)^k / (k-1)] = ((-1)^k) + 1 ?

Why? What else did you get when you converted the left-hand sides to common denominators and added?

Please show all of your steps. Thank you!

 

[DrPhil]

Hi,

this may seem silly but I can't figure out why these equations are equal...

1) Why is

(1)/(k+1) + (1)/(k-1) = (2k)/(k²-1) ?

2) Why is

[(-1)^k / (k+1)] + [(-1)^k / (k-1)] = ((-1)^k) + 1 ?

Thanks !

What have you tried? How far have you gotten?

In both of these the LCD is
......(k + 1)(k - 1) = ... (you can use the "product of sum and difference" rule)
To add the fractions, each has to be placed over the common denominator .. then add numerators.

For (2), treat it differently for even and odd k.

 

[JeffM]

Hi,

this may seem silly but I can't figure out why these equations are equal...

1) Why is

(1)/(k+1) + (1)/(k-1) = (2k)/(k²-1) ?

2) Why is

[(-1)^k / (k+1)] + [(-1)^k / (k-1)] = ((-1)^k) + 1 ?

Thanks !

Please check equation 2.

Suppose k = 4.

\(\displaystyle \dfrac{(-1)^4}{4 + 1} + \dfrac{(-1)^4}{4 - 1} = \dfrac{1}{5} + \dfrac{1}{3} = \dfrac{3}{15} + \dfrac{5}{15} = \dfrac{8}{15}.\)

\(\displaystyle (-1)^4 + 1 = 1 + 1 = 2 \ne \dfrac{8}{15}.\)

 

[lookagain]

Hi,

this may seem silly but I can't figure out why these equations are equal...

jones123, this is wrong phrasing. I thought from this sentence (before I read the rest of your post) that you were asking why those equations were equivalent to each other. But you din't mean to relate them together. Instead, you could have asked "Why are these equations valid (intending to consider each equation separately)?"