Geometric series Problem set A Worked solutions

Find each sum.

1. (a) 2 + 6 + 18 + … (8 terms)

   • \(S_n = \dfrac{u_1(r^n - 1)}{r - 1}\)
   • \(u_1 = 2,\ r = 3,\ n = 8:\ S_8 = \dfrac{2(3^8 - 1)}{3 - 1}\)
   • \(S_8 = 6560\)

2. (b) 5 − 10 + 20 − … (10 terms)

   • \(u_1 = 5,\ r = -10 \div 5 = -2,\ n = 10\)
   • \(S_{10} = \dfrac{5((-2)^{10} - 1)}{-2 - 1} = \dfrac{5(1024 - 1)}{-3}\)
   • \(S_{10} = -1705\)

3. (c) 100 + 50 + 25 + … (12 terms), to 3 significant figures

   • \(u_1 = 100,\ r = 0.5,\ n = 12\)
   • \(S_{12} = \dfrac{100(1 - 0.5^{12})}{1 - 0.5} = 199.95\ldots\)
   • \(S_{12} = 200\ (3\text{ s.f.})\)

A geometric series has common ratio 2, and the sum of its first 6 terms is 315. Find the first term.

• \(S_6 = \dfrac{u_1(2^6 - 1)}{2 - 1} = 63u_1 = 315\)
• \(u_1 = 315 \div 63 = 5\)

How many terms of 4 + 12 + 36 + … are needed for the sum to first exceed 10 000?

• \(u_1 = 4,\ r = 3:\ S_n = \dfrac{4(3^n - 1)}{2} = 2(3^n - 1)\)
• \(\text{Calculator table: } S_7 = 4372,\ S_8 = 13\,120\)
• n = 8 terms.

A message is sent to 4 people. Each of them sends it to 3 new people, and so on. How many people have received it after 6 rounds?

• \(\text{Round 1: 4 people, then} \times 3 \text{ each round: } u_1 = 4,\ r = 3\)
• \(S_6 = \dfrac{4(3^6 - 1)}{3 - 1} = \dfrac{4(729 - 1)}{2}\)
• 1456 people.

A starting salary is $28 000 and it rises by 3% each year. Find the total earned in the first 10 years.

• \(u_1 = 28\,000,\ r = 1.03,\ n = 10\)
• \(S_{10} = \dfrac{28\,000(1.03^{10} - 1)}{1.03 - 1} = 320\,988.62\ldots\)
• $320 988.62 (about $321 000).

A ball is dropped from a height of 5 m. After each bounce it rises to 60% of the height it fell from. Find the total distance the ball has travelled when it hits the ground for the 6th time.

• Rebounds before the 6th hit: 5. Heights: \(3,\ 1.8,\ \ldots\)\(\ (u_1 = 3,\ r = 0.6)\)
• \(\text{Up and down: } 2 \times \dfrac{3(1 - 0.6^{5})}{1 - 0.6} = 13.8336\)
• \(\text{Total: } 5 + 13.8336 = 18.8336\ldots\)
• About 18.8 m.

A challenge asks you to save £3 on day 1 and to save double the amount of the day before on each following day. On which day does the total first pass £10 000?

• \(u_1 = 3,\ r = 2\)
• \(S_n = 3(2^n - 1)\)
• \(S_{11} = £6141\) and \(S_{12} = £12 285\)
• The total first passes £10 000 on day 12.

The sum of the first \(n\) terms of a geometric series is \(S_n = 5(2^n - 1)\). Find the first term, the common ratio and the 6th term.

• \(u_1 = S_1 = 5(2^1 - 1) = 5\)
• \(u_2 = S_2 - S_1 = 5(2^2 - 1) - 5 = 15 - 5 = 10,\ r = 10 \div 5 = 2\)
• \(u_6 = S_6 - S_5 = 315 - 155 = 160\)
• \(\text{Check: } 5 \times 2^5 = 160\)