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IB Mathematics: Applications and Interpretation HL · 12.1: Sequences & Series · 12.1.7

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Infinite geometric series: the sum to infinity Problem set B

Show the first term and the ratio, check |r| < 1, then use the formula. Use the calculator to solve any quadratic.

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. Find the sum to infinity of each series.

    1. (a) \(\frac38+\frac{3}{40}+\frac{3}{200}+\dots\)
    2. (b) \(50+40+32+\dots\)
    3. (c) \(12-8+\frac{16}{3}-\frac{32}{9}+\dots\)

  2. Does the series \(10+11+12.1+\dots\) have a sum to infinity? Give a reason.

  3. Write \(0.\overline{54}\) as a fraction in lowest terms.

  4. Write each recurring decimal as a fraction in lowest terms.

    1. (a) \(0.\overline{5}\)
    2. (b) \(0.\overline{216}\)

  5. A geometric series has second term 4 and sum to infinity 20. Find the two possible values of the common ratio.

In context

  1. A charity receives 800 euros of donations in its first month. Each later month the donations are 75% of the month before, and this goes on for ever. Find the total donations received.

  2. A child on a swing moves 2.4 m on the first swing. Each swing is 95% as long as the swing before. Find the total distance travelled before the swing stops.

Stretch

  1. A geometric series has second term \(-8\) and sum to infinity 18. Find the common ratio and the first term.

  2. Write \(0.4\overline{27}\) as a fraction in lowest terms. Only the digits shown under the bar repeat.

jmaths.xyz · AI HL · 12.1.7 Infinite geometric series: the sum to infinity