Algebraic rules involving series: non-standard start and end indices

Sometimes, we know the formula \(F(n)\) for the summation of a sequence \(u_r\), with \(r\) starting from \(1\) and ending at \(n\). How can we use this formula to find the sum of the same sequence when the starting index is no longer \(1\)? The following expanded form provides an answer:

\[u_m + u_{m+1} + u_{m+2} + \dots + u_n\]

\[= \underbrace{u_1 + u_2 + \dots + u_{m-1}}_{\text{added in}} + u_m + u_{m+1} + \dots + u_n\]

\[- \underbrace{u_1 + u_2 + \dots + u_{m-1}}_{\text{take those added terms out}}\]

\[= \sum_{r=1}^{n} u_r - \sum_{r=1}^{m-1} u_r\]

\[= F(n) - F(m-1)\]

In short, if \(m\) and \(n\) are integers such that \(1 \le m \le n\) then

\[\sum_{r=m}^{n} u_r = \sum_{r=1}^{n} u_r - \sum_{r=1}^{m-1} u_r\]