Chapter 1: Sequences and Series

1.  State whether each geometric series is convergent or divergent.

a) 80 + 20 + 5 + \( \frac{5}{4} \) + …

b) –30 + 20 –  \( \frac{40}{3} \) + \( \frac{80}{9} \) – …

c) t₁ = –5,  r = \( \frac{1}{2} \)

d) t₁ = \( \frac{1}{3} \),  r = –2

2.  Determine the sum of each geometric series, if it exists.

a) t₁ = –4,  r = \( \frac{4}{5} \)

b) t₁ = 10,  r = \( \frac{-2}{3} \)

c) 10 + 10\( \sqrt[]{3} \) + 30 + 30\( \sqrt[]{3} \)+ …

d) \( \frac{5}{3} - \frac{5}{9} + \frac{5}{27} - \frac{5}{81} + ... \)

e) 8 + 8\( ( \frac{2}{3}) \)+ 8\( ( \frac{2}{3}) \)²+ 8\( ( \frac{2}{3}) \)³ + …

f)  – 2 – 2\( ( \frac{-3}{4}) \) – 2\( ( \frac{-3}{4}) \)² – 2\( ( \frac{-3}{4}) \)³ – …

3.  Express each of the following as an infinite geometric series. Determine the sum of the series.

a) 0.\( \vec{65} \)

b) 7.4\( \vec{5} \)

c) 0.123\( \vec{456} \)

4.  The general term of an infinite geometric series is t_n = 7\( ( \frac{1}{3}) \)^{n - 1}. Determine the sum of the series, if it exists.

5.  The sum of an infinite geometric series is \( \frac{10}{3} \) and the first term is 5. Determine the common ratio.

6.  The sum of an infinite geometric series is \( \frac{3 \pi }{2} \) and the common ratio is \( \frac{1}{2} \). Determine the first term.

7.  A ball is dropped from a height of 2.0 m onto a floor. On each bounce the ball rises to 75% of the height from which it fell. Calculate the total distance the ball travels before coming to rest.

8.  Determine the values of x such that the series 1 + x + x² + x³ + … has a sum.

9.  The sum of an infinite geometric series is three times the first term. Determine the common ratio.

10.  A new oil well produces 12 000 m³/month of oil. Its production is known to be dropping by 2.5% each month.

  a) What is the total production in the first year?

  b) Determine the total production of the well.