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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 A

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

Difficulty: first skills applying them stretch

Skills

  1. Find the sum to infinity of each series.

    1. (a) \(\frac34+\frac{3}{16}+\frac{3}{64}+\dots\)
    2. (b) \(18+12+8+\dots\)
    3. (c) \(10-5+2.5-1.25+\dots\)

  2. Does the series \(4+5+6.25+\dots\) have a sum to infinity? Give a reason.

  3. Write \(0.\overline{27}\) as a fraction in lowest terms.

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

    1. (a) \(0.\overline{8}\)
    2. (b) \(0.\overline{135}\)

  5. A geometric series has second term 6 and sum to infinity 25. Find the two possible values of the common ratio.

In context

  1. A trust pays a scholarship of 500 euros in the first year. Each later year the payment is 80% of the payment the year before, and this goes on for ever. Find the total the trust pays out.

  2. A pendulum swings through an arc of 40 cm on its first swing. Each swing is 90% as long as the swing before. Find the total distance the pendulum travels before it stops.

Stretch

  1. A geometric series has second term \(-6\) and sum to infinity 8. Find the common ratio and the first term.

  2. Write \(0.3\overline{18}\) 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