If T_{n} = \frac{ n-1 }{ n }

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

Question

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

anonymous · Answer

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

kinggeorge · Answer

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?

anonymous · Answer

n/n+1

kinggeorge · Answer

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

anonymous · Answer

n-2/n-1

kinggeorge · Answer

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.

anonymous · Answer

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

kinggeorge · Answer

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?

anonymous · Answer

just a sec

anonymous · Answer

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

anonymous · Answer

its ok i got the answer... i had a silly mistake

kinggeorge · Answer

We all make those from time to time.