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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.

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    You are making the mistake when assuming $a_n=a_1+(n-1)d_n$ – 2017-01-05
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    The fact that two differences yield a constant says that the formula for $a_n$ is quadratic in $n$. More generally, if $k$ differences yield a constant the formula for $a_n$ is of $k$th degree. The leading term is the constant divided by $k!$, so here the $n^2$ term should be $\frac 12n^2$. It is the same effect that integrating a constant $k$ times gives $\frac 1{k!}x^k$ – 2017-01-05
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    @RossMillikan correct. But how in the WORLD is this supposed to help the OP? – 2017-01-13
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    @bers: it gets the fact that the expression is quadratic and the quadratic term. Subtracting that off gives a liner expression which can be solved by inspection – 2017-01-13

2 Answers 2

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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.

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As others have already suggested in their comments and answer, your mistake lies in the step $a_n=a_1+(n-1)d_n$. Actually it should be replaced by $\boxed{a_n=a_1+\sum_\limits{i=1}^{n-1}d_i}$ where $n \ge2 $.

The rest of the solution will be as you have solved.