Geometric series Problem set B One question at a time

Find each sum.

1. (a) 3 + 12 + 48 + … (7 terms)

   • \(r = 4\)
   • \(S_n = \frac{u_1(r^n - 1)}{r - 1}\)
   • \(S_7 = \frac{3(4^7 - 1)}{4 - 1} = 16383\)

2. (b) 4 − 12 + 36 − … (6 terms)

   • \(r = -3\)
   • \(S_6 = \frac{4((-3)^6 - 1)}{-3 - 1} = \frac{4(728)}{-4}\)
   • \(S_6 = -728\)

3. (c) 160 + 80 + 40 + … (10 terms), to 3 significant figures

   • \(r = \tfrac{1}{2}\)
   • \(S_{10} = \frac{160(1 - 0.5^{10})}{1 - 0.5} = 319.6875\)
   • To 3 significant figures: 320

A geometric series has common ratio 3, and the sum of its first 5 terms is 484. Find the first term.

• \(S_5 = \frac{u_1(3^5 - 1)}{3 - 1} = 121u_1\)
• \(121u_1 = 484\)
• \(u_1 = 4\)

How many terms of 2 + 10 + 50 + … are needed for the sum to first exceed 20 000?

• \(S_n = \frac{2(5^n - 1)}{5 - 1} = \frac{5^n - 1}{2}\)
• \(S_6 = 7812,\ S_7 = 39\,062\)
• The sum first exceeds 20 000 at \(n = 7\).

A rumour reaches 3 people. Each of them passes it to 4 new people, and so on. How many people have heard it after 5 rounds?

• First round 3 people, each round 4 times as many: \(u_1 = 3,\ r = 4\)
• \(S_5 = \frac{3(4^5 - 1)}{4 - 1}\)
• \(S_5 = 1023\) people

A charity raises €12 000 in its first year, and each year it raises 5% more than the year before. Find the total raised in the first 9 years.

• \(u_1 = 12000,\ r = 1.05\)
• \(S_9 = \frac{12000(1.05^9 - 1)}{1.05 - 1}\)
• €132 318.77 (about €132 000)

A basketball is dropped from a height of 3 m. After each bounce it rises to 80% of the height it fell from. Find the total distance it has travelled when it hits the ground for the 5th time.

• Rebounds before the 5th hit: 4. Heights: \(2.4,\ 1.92,\ \ldots\)\(\ (u_1 = 2.4,\ r = 0.8)\)
• \(\text{Up and down: } 2 \times \dfrac{2.4(1 - 0.8^{4})}{1 - 0.8} = 14.1696\)
• \(\text{Total: } 3 + 14.1696 = 17.1696\ldots\)
• About 17.2 m.

A fund-raising page receives $2 on day 1. Each day it receives double the amount of the day before. On which day does the total raised first pass $50 000?

• \(u_1 = 2,\ r = 2\)
• \(S_n = 2(2^n - 1)\)
• \(S_{14} = $32 766\) and \(S_{15} = $65 534\)
• The total first passes $50 000 on day 15.

The sum of the first \(n\) terms of a geometric series is \(S_n = 3(4^n - 1)\). Find the first term, the common ratio and the 6th term.

• \(u_1 = S_1 = 3(4 - 1) = 9\)
• \(u_2 = S_2 - S_1 = 45 - 9 = 36,\ r = 36 \div 9 = 4\)
• \(u_6 = 9 \times 4^5 = 9216\)