The defining condition \(u_n - u_{n-1} = d\), where&nbsp;\(d\) 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&nbsp;\(u_n\) 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:
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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: \[
u_n = a + (n-1) d, \quad \qquad n \geq 1.
\]

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:
\[
u_n = a + (n-1) d, \quad \qquad n \geq 1.
\]

Method of difference: telescoping series

Method of difference: telescoping series