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IB Mathematics: Applications and Interpretation HL · 12.1: Sequences & Series · 12.1.10

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Sequences and series: revision Problem set B

Write down \(u_1\) and d (or \(u_1\) and r) first. Use the calculator for inequalities and equations in n.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. The sum of the first n terms of an arithmetic sequence is \(S_n=5n^2-2n\).

    1. (a) Write down the values of \(S_1\) and \(S_2\).
    2. (b) Write down the values of \(u_1\) and \(u_2\).
    3. (c) Find d, the common difference.
    4. (d) Find \(u_{15}\).
    5. (e) Find the greatest value of n for which \(u_n<200\).
    6. (f) Find the value of n for which \(S_n=952\).

  2. The term \(u_4\) of an arithmetic sequence is 61 and the common difference is \(-7\). Find \(u_1\).

  3. An arithmetic sequence has \(u_{4}=5\) and \(u_{12}=61\).

    1. (a) Find the common difference.
    2. (b) Find the first term.
    3. (c) Find \(u_{40}\).
    4. (d) Find \(S_{80}\).

  4. The term \(u_5\) of an arithmetic sequence is 26 and the common difference is \(-5\). The first and third terms of this sequence are the first two terms of a geometric sequence. Find the sixth term of the geometric sequence.

  5. The term \(u_3\) of an arithmetic sequence is 21 and the common difference is \(-5\). The first and fourth terms of this sequence are the first two terms of an infinite geometric series. Find the sum to infinity of the geometric series.

In context

  1. A student saves 15 euros in the first week. Each week after that the student saves 4 euros more than the week before.

    1. (a) Find the amount saved in week 30.
    2. (b) Find the total saved in the first 30 weeks.
    3. (c) The student wants to have saved at least 3000 euros. Find the least number of weeks needed.

  2. A runner runs 12 km in the first week of a training plan. Each week the runner runs 8% more than the week before. Give your answers to 3 significant figures.

    1. (a) Find the distance run in week 10.
    2. (b) Find the total distance run in the first 10 weeks.

  3. A forest team plants 72 trees in the first year. Each later year it plants 75% as many as the year before, and this continues for ever. Find the total number of trees planted.

Stretch

  1. An arithmetic sequence has \(S_{8}=104\) and \(S_{16}=400\). Find the first term and the common difference.

jmaths.xyz · AI HL · 12.1.10 Sequences and series: revision