Sequences: nth term and arithmetic series

Writing $3n$ (the common-difference part) without the constant, or writing the rule as an equation like $n=3n-2$.

Setting up the sum formula but mishandling the $(n-1)d$ term, or factorising the resulting quadratic in $n$ incorrectly.

Using the whole second difference as the $n^2$ coefficient instead of half of it — a second difference of $2$ means $an^2=1n^2$, not $2n^2$.

Using $n$ instead of $(n-1)$ in $S_n=\dfrac{n}{2}[2a+(n-1)d]$. With $a=7,d=4,n=20$ the bracket is $2(7)+19(4)=90$, giving $900$; using $20d$ overcounts.

The 3rd term means $n=3$, not the value $3$. For "the 3rd term is 13" write $a+2d=13$ (the 3rd term uses two lots of $d$), not $a+3d$.