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Sequences and Series

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Question 70

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

A laboratory is monitoring the mass of two newly synthesised isotopes, Alpha and Beta, during a stability experiment.

Initially, the mass of Isotope Alpha is 800 mg. A model predicts that the mass of Alpha will increase by 6% each year due to artificial enrichment, so that the mass at the end of each year forms a geometric sequence.

a.

Find, according to the model, the mass of Isotope Alpha 10 years after monitoring began. Give your answer to 3 significant figures.

[3]
b.

The mass of Isotope Beta is being monitored over the same period. Given that:

  • 4 years after monitoring began, the mass was 3200 mg
  • 9 years after monitoring began, the mass was 2100 mg
  • the mass of Isotope Beta at the end of each year is modelled as a geometric sequence

Find the equation of this model in the form

M=abt M = ab^t M=abt

where MMM is the mass of Isotope Beta, ttt years after monitoring began, and aaa and bbb are constants. Give the value of aaa and the value of bbb to 2 significant figures.

[4]
c.

At the time t=Tt = Tt=T, the mass of Isotope Alpha is equal to the mass of Isotope Beta.

Find, according to the models, the value of TTT, giving your answer to one decimal place.

[5]

Sequences and Series Questions

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