Write down the binomial expansion formula.

The binomial expansion for $(1+y)^n$ is given by:

In this problem, $y = \frac{1}{3}x$ and $n=8$.

Substitute $y = \frac{1}{3}x$ and $n=8$ into the formula:

Given: $\left(1+\frac{1}{3}x\right)^8 = 1 + \frac{2}{3}ax - \frac{4}{9}bx^2 + \dots$
Comparing the coefficient of $x$:

Comparing the coefficient of $x^2$:

The values of the constants are $\boxed{a=4, b=-7}$.

We are given the sum of the first $n$ terms of a sequence, $S_n = \frac{n}{2}(3n+1)$.
We need to find the sum of the terms from the $10^{th}$ term to the $30^{th}$ term. This can be calculated as $S_{30} - S_9$.

Substitute $n=30$ into the formula for $S_n$:

Substitute $n=9$ into the formula for $S_n$:

Sum = $S_{30} - S_9$
Sum = $1365 - 126$
Sum = 1239

The sum of the terms from the $10^{th}$ term to the $30^{th}$ term is $\boxed{1239}$.