1.2 Arithmetic sequences and seriesIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   An arithmetic sequence has first term $u_1=7$ and common difference $d=4$.

   (a)

   Find the value of $u_{20}$.

   [1 mark]

   • A$87$
   • B$83$
   • C$80$
   • D$79$

   (b)

   Find the sum of the first 20 terms of the sequence.

   [1 mark]

   • A$1800$
   • B$1660$
   • C$817$
   • D$900$

   (c)

   Find the value of $n$ for which $u_n=135$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cyclist trains for 20 days. On day 1 she cycles 12 km, and on each following day she cycles 1.5 km further than on the previous day.

   (a)

   Find the distance she cycles on day 15.

   [1 mark]

   • A$33$ km
   • B$34.5$ km
   • C$22.5$ km
   • D$31.5$ km

   (b)

   Find the total distance she cycles in the 20 days.

   [1 mark]

   • A$405$ km
   • B$810$ km
   • C$525$ km
   • D$484.5$ km

   (c)

   Find the first day on which she cycles more than 30 km.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An arithmetic sequence has $u_3=11$ and $u_8=31$.

   (a)

   Find the common difference $d$ and the first term $u_1$.

   [3 marks]

   (b)

   Using your GDC, find the value of $n$ for which $\sum_{k=1}^{n}u_k=1275$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A gym records its number of members at the start of each of the first five months: 210, 238, 267, 294 and 322. The manager models the number of members in month $n$ by an arithmetic sequence $u_n$. Each member pays a fee of 35 AED per month. Use your GDC where helpful.

   (a)

   (i) Use the data for months 1 and 5 to find an approximate common difference.
   (ii) Write down a formula for $u_n$.
   (iii) Use the model to predict the number of members in month 12.
   (iv) Comment on the reliability of this prediction.

   [6 marks]

   (b)

   (i) Find the total number of member-months in the first 12 months, according to the model.
   (ii) Hence find the total fees paid in the first 12 months.
   (iii) The gym needs total fees of at least 300 000 AED. Find the least number of months $n$ for which the model predicts this.

   [6 marks]

   Total for question 4: 12 marks

End of questions