We obviously cannot wait for million years to determine the half-life of technetium, or even years to determine the half-life of plutonium. The half-life of a substance is determined using a Geiger counter, which is essentially a tube filled with an inert gas. When a particle decays and produces radiation, it briefly ionizes the gas and produces a detectable electric pulse, which is amplified and counted by a detector (before digital displays were available, you would hear clicks and see a needle register on a dial). Determine the formula for the half-life of a substance with an initial count of pulses and a count of pulses minutes later.

The question is incomplete, the complete question is;

We obviously cannot wait for 4 million years to determine the half-life of technetium, or even 87.7 years to determine the half-life of plutonium. The half-life of a substance is determined using a Geiger counter, which is essentially a tube filled with an inert gas. When a particle decays and produces radiation, it briefly ionizes the gas and produces a detectable electric pulse, which is amplified and counted by a detector (before digital displays were available, you would hear clicks and see a needle register on a dial). Determine the formula for the half-life of a substance with an initial count of No pulses and a count of N1 pulses M minutes later.

M1/2 = 0.693/k

Explanation:

Given that the initial count rate is No pulses

The count rate after M minutes is N1

Then from;

A = Aoe^-kt

A= N1

Ao=No

I can now write;

N1 = Noe^-kM

The half life is the time taken for the activity of the radioactive nuclide to reach half its initial value. Hence;

N1/No =e^-kM = 1/2

Taking natural logarithm of both sides as shown below;

-kM1/2 = ln(1/2)

M1/2 = -  (ln(1/2)/k)

M1/2 = 0.693/k

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