Question

Proof Question

Answer 1

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

Answer 2

This is nice exercise for the hands

Answer 3

\[ T_{n+1} = \frac{n}{n+1}\]

Answer 4

So \[ T_{n+1} - T_{n-1} = \frac{ n }{ n+1 } - \frac{n-1}{n} \]

Answer 5

Sorry \[ T_{n+1} - T_{n-1} = \frac{ n }{ n+1 } - \frac{n-2}{n-1}\]

Answer 6

Just bring to the common denominator by multiplying the denominators and recall that
\[ (n+1)(n-1) = ...^2 - 1^2 \]

Answer 7

it seems like i cant go any further