Method of difference: telescoping series

The defining condition \(u_n - u_{n-1} = d\), where \(d\) is a constant, is a recurrence relation that generates the arithmetic sequence, once the seed -- the first term \(u_1\) -- is given. For us humans, a closed formula for \(u_n\) is convenient. Applying the defining condition repeatedly over these \(n-1\) positions \(2, 3, 4, \dots, n\):

\[ \begin{align*} u_2 - u_1 & = d \\ u_3 - u_2 & = d \\ u_4 - u_3 & = d \\ & \vdots \\ u_n - u_{n-1} & = d \\ \end{align*} \]

Adding all the \((n-1)\) equations, the "telescoping" terms are of opposite signs and get cancelled out:

Thus, we are left with: \(u_n - u_1 = (n-1)d\). So,
\[ u_n = u_1 + (n-1)d = a + (n-1)d. \]

The \(n \)th term of an arithmetic sequence \( \{ u_n\} \) whose first term is \(a\) and common difference \( d \) is calculated by the closed formula:
\[ u_n = a + (n-1) d, \quad \qquad n \geq 1. \]