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I have the sequence $\{0, 3, 8, 15, 24, 35,\dots\}$ and I need to find the explicit formula.

I am not sure how to go about doing this, but what I have started with is trying to find a way to express the changing variable getting added to the sequence. Each term adds $2$ plus the previous term. $n_1 + 3$, $n_2 + 5$,$n_3 + 7$ and so on. I am not entirely sure how to express this in a formula.

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If we add 1 to all the numbers in the sequence, we immediately see the squares, so the explicit form is $a(n)=n^2-1$ (with the first term having index 0).

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For the sequence, $$a_0=0, a_1=3, a_2=8, ...$$Your formula is $$a_n = n^2 + 2n,\text { n $\ge 0 $ }$$

You can prove it by induction.

If you want your sequence starts with $$a_1=0, a_2 =3, a_3 =8,...$$ Then the formula will be $$a_n = n^2 -1 , n\ge 1$$

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  • $\begingroup$ @Ovi For $n=3$ we get $a_3 =15$ which is correct base on starting with $a_0 =0, a_1=3, a_2=8, a_3 =15,...$ $\endgroup$ – Mohammad Riazi-Kermani Nov 17 '18 at 2:25
  • $\begingroup$ Oh sorry my mistake $\endgroup$ – Ovi Nov 17 '18 at 2:40
  • $\begingroup$ Thanks for the comment. I have both versions in my answer. $\endgroup$ – Mohammad Riazi-Kermani Nov 17 '18 at 2:42
  • $\begingroup$ Yes. In my mind when tried to compute $3^2$ I instead computed $2^3$ haha $\endgroup$ – Ovi Nov 17 '18 at 2:42
  • $\begingroup$ We all do these kind of mental math mistakes. $\endgroup$ – Mohammad Riazi-Kermani Nov 17 '18 at 2:44

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