# How do you find the nth term formula 3,8,15,24,... ?

Jun 13, 2016

$a \left(n\right) = a \left(n - 1\right) + 2 \cdot \left(n + 1\right) + 1$

#### Explanation:

Having the first term of the sequence
$\text{ }$
$a \left(0\right) = 3$
$\text{ }$
$a \left(1\right) = 3 + 5 = 8$
$\text{ }$
We realized that
$\text{ }$
$a \left(1\right) = a \left(0\right) + 2 \cdot 2 + 1$

We also have :
$\text{ }$
$a \left(2\right) = a \left(1\right) + 2 \cdot 3 + 1 = 8 + 7 = 15$
$\text{ }$
$a \left(3\right) = a \left(2\right) + 2 \cdot 4 + 1 = 15 + 9 = 24$

From above we can realize that each term is the sum of the previous
$\text{ }$
term and 2*(sequence coefficient added to 1) and 1
$\text{ }$
So the nth term will be:
$\text{ }$
$a \left(n\right) = a \left(n - 1\right) + 2 \cdot \left(n + 1\right) + 1$