Recognise factorial growth from the recurrence

Successive multiplication gives:

For example $u_5=5u_4=120$, $u_6=6u_5=720$, $u_7=7u_6=5040$. Some source symbolic lines insert stray $+1$ terms and use $u_3$ in the $u_5$ line; the stated recurrence, numerical products and calculator table consistently give the values above.

$u_n=n(n-1)\cdots2\cdot1=n!$. The source also obtains the same first seven values from a calculator recurrence table and recognises $1!,2!,\ldots,7!$. Induction confirms the pattern: from $u_n=n!$, $u_{n+1}=(n+1)n!=(n+1)!$, with base $u_1=1!$.