Missing Numbers - 4

Explicit formula:

$\begin{aligned} a_n & = \frac 12 n^2 + \frac 32 n + 3 \\ \Rightarrow a_6 & = 30 \end{aligned}$

$5+3=8$ $8+4=12$ $12+5=17$ $17+6=23$ $23+7=30$

Generally, $a_{n+1}=a_n+n+2$

The sequence was adding n first, and then adding ascending amounts: n+1, n+2, n+3, and n+4. It started out by adding 3 to 5. Then, it added four. It kept on going until it reached thirty. 3+4+5+6+7=25. 25+5 is 30.

$The\ logic\ is\ 5 + 3 = 8, 8 + 4 = 12, 12 + 5 = 17, 17 + 6 = 23, 23 + 7 =$ $\boxed{30}$ $...$