Sequences, term-to-term and position-to-term rules

Two ways to describe a sequence

The term-to-term rule says how to get the next term from the one before: $3,7,11,15,\ldots$ has the rule “add $4$”. The position-to-term rule (the $n$th term) gives any term straight from its position: here it is $4n-1$.

Finding the $n$th term of a linear sequence

Step one: find the common difference $d$ — that is the coefficient of $n$. Step two: work out the constant by testing $n=1$. For $3,7,11,15$: $d=4$, so start with $4n$; at $n=1$ that gives $4$ but we want $3$, so subtract $1$. The rule is $4n-1$.

Why the $n$th term matters

The term-to-term rule cannot answer “what is the $100$th term?” without listing a hundred numbers. The $n$th term answers it in one line: $4(100)-1=399$. That is exactly the kind of question Checkpoint asks.