Mixed practice: which model? Problem set B Worked solutions

Say whether each sequence is arithmetic, geometric or neither.

1. (a) 18, 14.5, 11, 7.5, …

   • Differences: \(14.5 - 18 = -3.5,\ 11 - 14.5 = -3.5,\ 7.5 - 11 = -3.5\)
   • Constant difference: arithmetic.

2. (b) 16, 20, 25, 31.25, …

   • Ratios: \(20 \div 16 = 1.25,\ 25 \div 20 = 1.25,\ 31.25 \div 25 = 1.25\)
   • Constant ratio: geometric.

3. (c) 1, 3, 6, 10, …

   • Differences: 2, 3, 4. Ratios: 3, 2, 1.67…
   • Neither is constant, so neither.

4. (d) 48, −24, 12, −6, …

   • Ratios: \(-24 \div 48 = -\frac{1}{2},\ 12 \div (-24) = -\frac{1}{2},\ -6 \div 12 = -\frac{1}{2}\)
   • Constant ratio: geometric.

An arithmetic sequence has \(u_1 = 8,\ d = 6\). A geometric sequence has \(v_1 = 8,\ r = 1.25\). Find the smallest \(n > 1\) for which \(v_n > u_n\).

• \(u_n = 8 + 6(n-1),\ v_n = 8 \times 1.25^{n-1}\)
• \(n = 10: u_{10} = 62,\ v_{10} = 59.6\)
• \(n = 11: u_{11} = 68,\ v_{11} = 74.5\)
• \(n = 11\)

€4500 is invested at 3.2% simple interest per year. Find the amount after 7 years.

• Interest each year: \(4500 \times 0.032 = 144\)
• \(4500 + 7 \times 144 = 5508\)
• The amount is €5508.

Two warehouse rental agreements both cost $400 in the first month. Agreement A goes up by $30 each month. Agreement B goes up by 5% each month. In which month does agreement B first cost more than agreement A?

• \(A_n = 400 + 30(n-1),\ B_n = 400 \times 1.05^{n-1}\)
• Month 17: A = $880.00, B = $873.15
• Month 18: A = $910.00, B = $916.81
• Month 18

A puppy's mass is measured each month: 4.0, 5.4, 7.1, 8.4, 10.0 kg. Model the masses as an arithmetic sequence and predict the mass in month 12.

• \(d = \frac{u_5 - u_1}{4} = \frac{10.0 - 4.0}{4} = 1.5\)
• \(u_{12} = 4 + 11(1.5)\)
• \(u_{12} = 20.5\) kg

A cyclist rides 12 km on day 1 and 3 km more each day. (a) How many km does she ride on day 18? (b) How many km does she ride in total over the 18 days? Say which is a term and which is a sum.

• (a) A term: \(u_{18} = 12 + 17(3) = 63\) km
• (b) A sum: \(S_{18} = \frac{18}{2}(12 + 63)\)
• \(S_{18} = 675\) km

A pattern is made of circles. Stage 1 has 3 circles, stage 2 has 6 circles and stage 3 has 12 circles. (a) Is the number of circles arithmetic or geometric? Give the common difference or ratio. (b) How many circles are in stage 9? (c) How many circles are there in stages 1 to 9 altogether?

• \(3,\ 6,\ 12,\ \ldots\)
• (a) Each stage is 2 times the one before: geometric with \(u_1 = 3,\ r = 2\).
• \((b)\ u_{9} = 3 \times 2^{8} = 768\)
• \((c)\ S_{9} = \dfrac{3(2^{9} - 1)}{2 - 1} = 1533\)