1.11 Sum of infinite geometric sequencesIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A geometric sequence has first term $u_1=12$ and common ratio $r=\frac13$.

   (a)

   Find the sum to infinity of the sequence.

   [1 mark]

   • A$9$
   • B$36$
   • C$4$
   • D$18$

   (b)

   Find $u_4$.

   [1 mark]

   • A$\frac{4}{27}$
   • B$\frac43$
   • C$\frac49$
   • D$4$

   (c)

   Use your GDC to find the least value of $n$ for which the sum of the first $n$ terms is greater than $17.9$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A ball is dropped from a height of $2$ m onto a hard floor. After each bounce it rises to $80\%$ of the height from which it last fell.

   (a)

   Find the height to which the ball rises after the third bounce.

   [1 mark]

   • A$1.28$ m
   • B$1.024$ m
   • C$0.8192$ m
   • D$1.6$ m

   (b)

   Find the total distance travelled by the ball until it comes to rest.

   [1 mark]

   • A$18$ m
   • B$10$ m
   • C$16$ m
   • D$20$ m

   (c)

   Use your GDC to find the number of the first bounce after which the ball rises to less than $0.1$ m.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The recurring decimal $0.272727\ldots$ can be written as the sum of an infinite geometric series.

   (a)

   Write down the first term and the common ratio of the series $0.27+0.0027+0.000027+\ldots$, and explain why its sum to infinity exists.

   [3 marks]

   (b)

   Find the exact value of $0.272727\ldots$ as a fraction in its simplest form.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A patient takes a $200$ mg dose of a drug each morning. During each following $24$ hours the body removes $30\%$ of the drug present, so $70\%$ remains just before the next dose. A GDC may be used.

   (a)

   (i) Find the amount of the drug in the patient's body just after the third dose.
   (ii) Show that the amount just after the $n$th dose is $\frac{2000}{3}\left(1-0.7^{n}\right)$ mg.
   (iii) Find the amount that the drug level approaches in the long term. Give your answer to 3 significant figures.

   [6 marks]

   (b)

   (i) The level is unsafe once the amount just after a dose exceeds $600$ mg. Find the number of the first dose after which the level is unsafe.
   (ii) Explain why the amount just after a dose never exceeds $700$ mg.

   [6 marks]

   Total for question 4: 12 marks

End of questions