What's wrong with this sequential reasoning for a formula for a_n? The following sequence is given. Write a formula for $a_n$.
$\displaystyle\{1,2,4,7,11,16\}$
Note the arithmetic formula.
$\displaystyle a_n=a_1+\left(n-1\right)d$
Define $d$ as $d_n$, where $d_n$ is the difference between term $n$ and its succeeding term, $n+1$.
$\displaystyle a_n=a_1+\left(n-1\right)d_n$
$\displaystyle d_n=d_1+\left(n-1\right)g$
Where $g$ is the common difference for $d_n$.
$\displaystyle a_n=a_1+\left(n-1\right)\left(d_1+\left(n-1\right)g\right)$
$a_n\in\{1,2,4,7,11,16\}$
$d_n\in\{1,2,3,4,5\}$
$g=1$
$\displaystyle a_n=a_1+\left(n-1\right)\left(d_1+\left(n-1\right)\right)$
$\displaystyle a_n=1+\left(n-1\right)\left(1+\left(n-1\right)\right)$
$\displaystyle a_n=n^2-n+1$
But this doesn't yield the correct sequence.

Following your notations :- 
$d_n$ is the difference between the $(n+1)th$ and $nth$ term $\implies d_n=a_{n+1}-a_n$.
Also, $d_n=d_1+(n-1)g$ and $d_1=1$ and $g=1\implies d_n=n$.
Thus $a_{n+1}-a_n=n\implies a_{n+1}=a_n+n, \ n\geq 1$.
The mistake you were making is to assume that $a_n=a_1+(n-1)d$, which is true with a fixed common difference.
