# Explicit formula for the nth term of the sequence

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.

## 2 Answers

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).

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 $$a_n = n^2 -1 , n\ge 1$$

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