For the (T)-period moving average forecast method, show that [ begin{aligned} f_{n+1} & =frac{sum_{t=n-T+1}^{n} d_{t}}{T} & =f_{n}+left[frac{d_{n}-d_{n-T}}{T} ight] end{aligned} ]

For the \(T\)-period moving average forecast method, show that

\[ \begin{aligned} f_{n+1} & =\frac{\sum_{t=n-T+1}^{n} d_{t}}{T} \\ & =f_{n}+\left[\frac{d_{n}-d_{n-T}}{T}\right] \end{aligned} \]