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Step 1: Understand that the decay rate of a radioactive element decreases over time according to the formula:
$$ N(t) = N_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} $$ where
- $N(t)$ is the amount remaining after time $t$,
- $N_0$ is the initial amount,
- $T_{1/2}$ is the half-life, and
- $t$ is the elapsed time.

Step 2: Given the initial decay rate $N_0 = 10^3$ disintegration/sec and $T_{1/2} = 1$ second.
Step 3: For $t = 1$ second:
$$ N(1) = 10^3 \cdot \left(\frac{1}{2}\right)^{\frac{1}{1}} = 10^3 \cdot \frac{1}{2} = 500 \text{ disintegration/sec} $$

Step 4: For $t = 3$ seconds:
$$ N(3) = 10^3 \cdot \left(\frac{1}{2}\right)^{\frac{3}{1}} = 10^3 \cdot \left(\frac{1}{2}\right)^{3} = 10^3 \cdot \frac{1}{8} = 125 \text{ disintegration/sec} $$

Step 5: Therefore, after 1 second the decay rate is 500 disintegration/sec and after 3 seconds it is 125 disintegration/sec.