Infinite geometric series: the sum to infinity Problem set B One question at a time

Find the sum to infinity of each series.

1. (a) \(\frac38+\frac{3}{40}+\frac{3}{200}+\dots\)

   • \(u_1=\frac{3}{8},\ r=\frac{1}{5}\)
   • \(|r|<1\), so \(S_\infty=\frac{\frac{3}{8}}{1-\frac{1}{5}}\)
   • \(S_\infty=\frac{15}{32}\)

2. (b) \(50+40+32+\dots\)

   • \(u_1=50,\ r=\frac{4}{5}\)
   • \(|r|<1\), so \(S_\infty=\frac{50}{1-\frac{4}{5}}\)
   • \(S_\infty=250\)

3. (c) \(12-8+\frac{16}{3}-\frac{32}{9}+\dots\)

   • \(u_1=12,\ r=-\frac{2}{3}\)
   • \(|r|<1\), so \(S_\infty=\frac{12}{1-(-\frac{2}{3})}\)
   • \(S_\infty=\frac{36}{5}\)

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

• \(r=\frac{11}{10}=1.1\)
• \(\frac{\frac{121}{10}}{11}=1.1\), so the series is geometric
• \(|r|>1\): the terms grow, so there is no sum to infinity

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

• \(0.\overline{54}=54/100+54/10\,000+\dots\), so \(u_1=\frac{54}{100},\ r=\frac{1}{100}\)
• \(S_\infty=\frac{54/100}{99/100}=\frac{54}{99}\)
• Lowest terms: \(\frac{6}{11}\)

Write each recurring decimal as a fraction in lowest terms.

1. (a) \(0.\overline{5}\)

   • \(u_1=\frac{5}{10},\ r=\frac1{10}\)
   • \(S_\infty=\frac{5/10}{9/10}=\frac{5}{9}\)

2. (b) \(0.\overline{216}\)

   • \(u_1=\frac{216}{1000},\ r=\frac{1}{1000}\)
   • \(S_\infty=\frac{216}{999}\)
   • Lowest terms: \(\frac{8}{37}\)

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

• \(u_1=\frac{4}{r}\), so \(\frac{4/r}{1-r}=20\)
• \(4=20r(1-r)\), so \(20r^2-20r+4=0\)
• From the calculator: \(r=0.276\) or \(r=0.724\)
• Both have \(|r|<1\), so both are allowed

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.

• A geometric series: \(u_1=800,\ r=0.75\)
• \(|r|<1\), so \(S_\infty=\frac{800}{1-0.75}\)
• \(S_\infty=3200\)

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.

• A geometric series: \(u_1=2.4,\ r=0.95\)
• \(S_\infty=\frac{2.4}{1-0.95}\)
• \(S_\infty=48\) m

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

• \(u_1=\frac{-8}{r}\), so \(-8=18r(1-r)\) and \(18r^2-18r-8=0\)
• From the calculator: \(r=-\frac{1}{3}\) or \(r=\frac{4}{3}\)
• \(|\frac{4}{3}|>1\), so reject it. \(r=-\frac{1}{3}\)
• \(u_1=\frac{-8}{-\frac{1}{3}}=24\)

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

• \(0.4\overline{27}=0.4+0.027+0.00027+\dots\)
• The series after 0.4 has \(u_1=0.027=\frac{27}{1000},\ r=\frac1{100}\)
• \(S_\infty=\frac{27/1000}{99/100}=\frac{27}{990}=\frac3{110}\)
• \(\frac4{10}+\frac3{110}=\frac{44}{110}+\frac{3}{110}=\frac{47}{110}\)