Shorter half-life and therefore very radioactive -- why?

[CPW]

TL;DR

In reading through The Physics of Energy, the textbook describes the decay chain of U-238:
"The longest half-life of any descendent in the chain is less 1 million years. Many half-lives are much shorter, making those nuclides very radioactive."

Why does having a short half-life make a radionuclide very radioactive?

In reading through The Physics of Energy, the textbook describes the decay chain of U-238:
"The longest half-life of any descendent in the chain is less 1 million years. Many half-lives are much shorter, making those nuclides very radioactive."
Why does having a short half-life make a radionuclide very radioactive?

My answer, qualitatively:
Relative to the time available for particle emissions from the long-lived parent radionuclide (U-238), the short-lived descendants have much less time to perform all the necessary particle emissions. And therefore, the short-lived radionuclides will have much higher radioactivity, as they will be emitting particles more frequently.
(Am I correct?)

However, quantitatively, I'm stuck.
I'd like a more formal answer than my answer above.

The amount of radioactivity (Bq) must be related to the number of disintegrations per gram per second.
But is there an equation relating these quantities?

 

[dRic2]

The activity ($A$), which is the number of disintegration per unit time, is given by
$$A = \lambda N$$
where $\lambda$ is the decay-constant and $N$ is the number of particles in the sample. If you assume that the number of particles $N$ in the sample does not change significantly during the period of time in which you measure the radioactivity, then you see that the higher the $\lambda$, the higher the activity (number of disintegrations). It also turns out that the decay-constant $\lambda$ and the half-life $\tau_{1/2}$ are related by:
$$\tau_{1/2} = \frac {\ln 2} {\lambda}$$.
To summarize, small half-life -> big decay-constant ->big number of disintegrations per seconds = high activity.

 

[CPW]

Thank you.

 

[DrDu]

I just want to add that the intermediate nuclides are higher radioactive in the sense that the specific activity, i.e. the activity per gram of the nuclide is higher. However, if the decay products are in equilibrium with each other, the activity of all isotopes is the same, irrespective of their half live.