Geometric sequences Problem set A One question at a time

Find the term asked for.

1. (a) \(u_1 = 3,\ r = 2\). Find \(u_{10}\).

   • \(u_n = u_1 r^{n-1}\)
   • \(u_{10} = 3 \times 2^9\)
   • \(u_{10} = 1536\)

2. (b) 4, −12, 36, … Find \(u_7\).

   • \(r = -12 \div 4 = -3\)
   • \(u_7 = 4 \times (-3)^6\)
   • \(u_7 = 2916\)

3. (c) 80, 40, 20, … Find \(u_9\).

   • \(r = 40 \div 80 = 0.5\)
   • \(u_9 = 80 \times 0.5^8\)
   • \(u_9 = 0.3125\)

Find the common ratio and the first term.

1. (a) \(u_2 = 6,\ u_5 = 162\)

   • \(u_5 / u_2 = r^3 = \frac{162}{6} = 27\)
   • \(r = 3\)
   • \(u_1 = 6 \div 3 = 2\)

2. (b) \(u_3 = 20,\ u_5 = 80\)

   • \(u_5 / u_3 = r^2 = \frac{80}{20} = 4\)
   • \(r = 2 \text{ or } r = -2\)
   • \(u_1 = 20 \div r^2 = 20 \div 4 = 5\ (\text{both cases})\)

Write an expression for \(u_n\).

1. (a) 5, 15, 45, …

   • \(u_1 = 5,\ r = 15 \div 5 = 3\)
   • \(u_n = 5 \times 3^{n-1}\)

2. (b) 64, 32, 16, …

   • \(u_1 = 64,\ r = 32 \div 64 = \tfrac{1}{2}\)
   • \(u_n = 64 \times \left(\tfrac{1}{2}\right)^{n-1}\)

Answer each part.

1. (a) How many terms are in the sequence 7, 14, 28, …, 3584?

   • \(u_1 = 7,\ r = 2:\ 7 \times 2^{n-1} = 3584\)
   • \(2^{n-1} = 512 = 2^9\)
   • \(n - 1 = 9 \Rightarrow n = 10\)

2. (b) Find the first term of 3, 12, 48, … that is greater than 10 000.

   • \(u_1 = 3,\ r = 4:\ u_n = 3 \times 4^{n-1}\)
   • \(\text{Calculator table: } u_6 = 3072,\ u_7 = 12\,288\)
   • The 7th term, 12 288, is the first above 10 000.

A ball is dropped from 2 m. After each bounce it rises to 70% of the height before. Find the height it reaches after the 5th bounce.

• \(\text{Height after } n \text{ bounces} = 2 \times 0.7^n \quad (u_1 = 2,\ r = 0.7)\)
• \(2 \times 0.7^5 = 0.33614\ldots\)
• 0.336 m.

A town of 42 000 people grows by 2.5% each year. (a) Estimate the population after 8 years. (b) After how many whole years does it first pass 60 000?

• \(\text{Population after } n \text{ years} = 42\,000 \times 1.025^n \quad (r = 1.025)\)
• \((a)\ 42\,000 \times 1.025^8 = 51\,172.92\ldots \approx 51\,200\)
• \((b)\ \text{Calculator table: } n = 14: 59\,345\ldots,\ n = 15: 60\,829\ldots\)
• (a) About 51 200. (b) 15 years.

A van bought for £18 000 loses 12% of its value each year. Find its value after 6 years.

• \(\text{Loses 12\%: } r = 1 - 0.12 = 0.88\)
• \(18\,000 \times 0.88^6 = 8359.27\ldots\)
• £8 359.27 (about £8360).

A car is bought for £25 000 and loses 15% of its value each year. (a) Find its value after 3 years. (b) After how many complete years is it first worth less than half of its original price?

• \(u_n = 25000 \times 0.85^{n}\)
• \((a)\ 25000 \times 0.85^{3} = 15 353.12\)
• \((b)\ 25000 \times 0.85^n < 12500 \Rightarrow n > 4.27\ldots\)
• After 4 years it is worth £13 050.16; after 5 years £11 092.63. So 5 complete years.

\(x,\ x + 4\) and \(x + 12\) are three consecutive terms of a geometric sequence. Find \(x\) and the common ratio.

• \(\dfrac{x + 4}{x} = \dfrac{x + 12}{x + 4}\)
• \((x + 4)^2 = x(x + 12)\)
• \(x^2 + 8x + 16 = x^2 + 12x\)
• \(16 = 4x \Rightarrow x = 4\)
• Terms 4, 8, 16; r = 2.