Geometric sequences Problem set B Worked solutions

Find the term asked for.

1. (a) \(u_1 = 4,\ r = 3\). Find \(u_8\).

   • \(u_n = u_1 r^{n-1}\)
   • \(u_8 = 4 \times 3^7 = 8748\)

2. (b) 7, −14, 28, … Find \(u_7\).

   • \(r = -14 \div 7 = -2\)
   • \(u_7 = 7 \times (-2)^6 = 448\)

3. (c) 72, 36, 18, … Find \(u_7\).

   • \(r = 36 \div 72 = \tfrac{1}{2}\)
   • \(u_7 = 72 \times \left(\tfrac{1}{2}\right)^6 = 1.125\)

Find the common ratio and the first term.

1. (a) \(u_2 = 12,\ u_5 = 768\)

   • \(r^3 = 768 \div 12 = 64\)
   • \(r = 4\)
   • \(u_1 = 12 \div 4 = 3\)

2. (b) \(u_3 = 18,\ u_5 = 162\)

   • \(r^2 = 162 \div 18 = 9\)
   • \(r = 3\) or \(r = -3\)
   • \(u_1 = 18 \div 9 = 2\) in both cases

Write an expression for \(u_n\).

1. (a) 2, 10, 50, …

   • \(u_n = u_1 r^{n-1}\)
   • \(u_1 = 2,\ r = 5\)
   • \(u_n = 2 \times 5^{n-1}\)

2. (b) 81, 27, 9, …

   • \(u_n = u_1 r^{n-1}\)
   • \(u_1 = 81,\ r = \tfrac{1}{3}\)
   • \(u_n = 81 \times \left(\tfrac{1}{3}\right)^{n-1}\)

Answer each part.

1. (a) How many terms are in the sequence 5, 15, 45, …, 10 935?

   • \(5 \times 3^{n-1} = 10935\)
   • \(3^{n-1} = 2187 = 3^7\)
   • \(n = 8\)

2. (b) Find the first term of 2, 6, 18, … that is greater than 10 000.

   • \(2 \times 3^{n-1} > 10000\)
   • \(u_8 = 4374,\ u_9 = 13\,122\)
   • The 9th term, 13 122

A basketball is dropped from 5 m. After each bounce it rises to 60% of the height before. Find the height it reaches after the 4th bounce.

• The height is multiplied by 0.6 at each bounce.
• \(5 \times 0.6^4\)
• \(= 0.648\) m

A village of 6500 people grows by 3.5% each year. (a) Estimate the population after 9 years. (b) After how many whole years does it first pass 10 000?

• \(6500 \times 1.035^9 = 8858.8\)
• (a) About 8860 people
• \(6500 \times 1.035^{12} = 9821.9\) and \(6500 \times 1.035^{13} = 10\,165.7\)
• (b) 13 years

A tractor bought for $45 000 loses 15% of its value each year. Find its value after 5 years.

• Each year the value is multiplied by 0.85.
• \(45000 \times 0.85^5\)
• $19 966.74 (about $20 000)

A machine is bought for $60 000 and loses 18% of its value each year. (a) Find its value after 2 years. (b) After how many complete years is it first worth less than half of its original price?

• \(u_n = 60000 \times 0.82^{n}\)
• \((a)\ 60000 \times 0.82^{2} = 40 344.00\)
• \((b)\ 60000 \times 0.82^n < 30000 \Rightarrow n > 3.49\ldots\)
• After 3 years it is worth $33 082.08; after 4 years $27 127.31. So 4 complete years.

\(x,\ x + 6\) and \(x + 24\) are three consecutive terms of a geometric sequence. Find \(x\) and the common ratio.

• \(\dfrac{x+6}{x} = \dfrac{x+24}{x+6}\)
• \(x^2 + 12x + 36 = x^2 + 24x\)
• \(12x = 36,\ x = 3\)
• The terms are 3, 9, 27 and \(r = 3\).