1.3 Geometric sequences and seriesIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A geometric sequence has first term $u_1=4$ and common ratio $r=3$.

   (a)

   Find the value of $u_6$.

   [1 mark]

   • A$2916$
   • B$324$
   • C$19$
   • D$972$

   (b)

   Find the sum of the first 6 terms.

   [1 mark]

   • A$484$
   • B$728$
   • C$1456$
   • D$4372$

   (c)

   Find the least value of $n$ for which $u_n>10\,000$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A virus spreads so that the number of new cases each day is 1.5 times the number of new cases on the day before. On day 1 there are 40 new cases.

   (a)

   Find the number of new cases on day 6, to the nearest whole number.

   [1 mark]

   • A$456$
   • B$304$
   • C$300$
   • D$47.6$

   (b)

   Find the total number of new cases in the first 8 days, to the nearest whole number.

   [1 mark]

   • A$1970$
   • B$2995$
   • C$985$
   • D$493$

   (c)

   Find the first day on which the number of new cases is greater than 1000.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A geometric sequence has $u_2=12$ and $u_5=96$.

   (a)

   Find the common ratio $r$ and the first term $u_1$.

   [3 marks]

   (b)

   Find the least value of $n$ for which $\sum_{k=1}^{n}u_k>5000$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Aisha and Omar both start a job in year 1 on a salary of 36 000 AED. Aisha's salary increases by 4% at the start of each following year. Omar's salary increases by 1 600 AED at the start of each following year. Use your GDC where helpful.

   (a)

   (i) Write down the common ratio of Aisha's yearly salaries.
   (ii) Find Aisha's salary in year 10.
   (iii) Find Aisha's total earnings over the first 10 years.
   (iv) Find Aisha's mean annual salary over the first 10 years.
   Give money answers to 2 decimal places.

   [6 marks]

   (b)

   (i) Write down a formula for Omar's salary in year $n$.
   (ii) Find the first year in which Aisha's salary is greater than Omar's salary.
   (iii) Find Omar's total earnings over the first 10 years. Hence state, with a reason, who earns more in total over the 10 years.

   [6 marks]

   Total for question 4: 12 marks

End of questions