Question

(4,7,12,19) How do you right this in recursive and explicit equations??????? pls help

Answer 1

You can see how the difference between two consecutive terms is constantly increasing:

$a_1 = 4$

$a_2 = 7 = 4+3$

$a_3 = 12 = 7+5$

$a_4 = 19 = 12+7$

So, for the next terms we'll have to add +9, +11, +13 and so on.

Also, note that $a_2$ is obtained by adding the 2nd odd number to $a_1$, $a_3$ is obtained by adding the 3rd odd number to $a_2$, and so on.

So, the recursive formula is

$a_n = a_{n-1}+(2n-1),\quad a_1=1$

For the explicit formula, recall that the sum of the first n odd numbers is n squared. Taking into account the fact that we're not starting from 1, we have

$a_n = n^2+3$