## aceace 2 years ago If T_{n} = \frac{ n-1 }{ n } , prove T_{n+1} - T _{n-1} = \frac{ 2 }{ n^2 -1 }

1. aceace

If T_{n} = \frac{ n-1 }{ n }, prove T_{n+1} - T _{n-1} = \frac{ 2 }{ n^2 -1 }

2. KingGeorge

First, you want to write out what $$T_{n+1}$$ and $$T_{n-1}$$ are using the fact that $$\displaystyle T_n=\frac{n-1}{n}$$. Can you tell me what $$T_{n+1}$$ is?

3. aceace

n/n+1

4. KingGeorge

Right, and $$T_{n-1}$$?

5. aceace

n-2/n-1

6. KingGeorge

Bingo. So you have $T_{n+1}-T_{n-1}=\frac{n}{n+1}-\frac{n-2}{n-1}$Find a common denominator, and simplify.

7. aceace

yeh i did that and got to the answer 2/(1-n) which is wrong

8. KingGeorge

Alright, to get a common denominator, you do the following, and simplify. $\left(\frac{n}{n+1}\cdot\frac{n-1}{n-1}\right)-\left(\frac{n-2}{n-1}\cdot\frac{n+1}{n+1}\right)$ Does this help?

9. aceace

just a sec

10. aceace

yeh i got that step except i cant get the final answer

11. aceace