Mixed practice: which model? Problem set A Worked solutions

Say whether each sequence is arithmetic, geometric or neither.

1. (a) 12, 9.5, 7, 4.5, …

   • \(9.5 - 12 = -2.5,\ 7 - 9.5 = -2.5,\ 4.5 - 7 = -2.5\)
   • Common difference: arithmetic, d = −2.5.

2. (b) 8, 12, 18, 27, …

   • \(12 \div 8 = 1.5,\ 18 \div 12 = 1.5,\ 27 \div 18 = 1.5\)
   • Common ratio: geometric, r = 1.5.

3. (c) 1, 4, 9, 16, …

   • \(\text{Differences: } 3, 5, 7;\ \text{ratios: } 4, 2.25, 1.78\)
   • Neither is constant: neither.

4. (d) 54, −18, 6, −2, …

   • \(-18 \div 54 = -\tfrac{1}{3},\ 6 \div (-18) = -\tfrac{1}{3},\ -2 \div 6 = -\tfrac{1}{3}\)
   • \(\text{Common ratio: geometric, } r = -\tfrac{1}{3}\)

An arithmetic sequence has \(u_1 = 5,\ d = 3\). A geometric sequence has \(v_1 = 5,\ r = 1.2\). Find the smallest \(n > 1\) for which \(v_n > u_n\).

• \(u_n = 5 + 3(n - 1) = 3n + 2,\quad v_n = 5 \times 1.2^{n-1}\)
• \(\text{Calculator table: } n = 12: u_{12} = 38,\ v_{12} = 37.15\)
• \(n = 13: u_{13} = 41,\ v_{13} = 44.58\)
• n = 13.

£3500 is invested at 4.2% simple interest per year. Find the amount after 5 years.

• \(\text{Simple interest: the same amount is added each year}\)
• \(3500 \times 4.2\% = 147 \text{ a year}\)
• 3500 + 5 × 147 = 4235, so £4235.

Two phone plans both cost £500 in the first month of a contract. Plan A goes up by £40 each month. Plan B goes up by 5% each month. In which month does plan B first cost more than plan A?

• \(A:\ 500 + 40(n - 1) \text{ (arithmetic)},\quad B:\ 500 \times 1.05^{n-1} \text{ (geometric)}\)
• \(\text{Calculator table: month } 19:\ A = 1220,\ B = 1203.31\)
• \(\text{Month } 20:\ A = 1260,\ B = 1263.48\)
• Month 20.

A plant's height is measured each week: 12.0, 14.1, 15.9, 18.2, 20.0 cm. Model the heights as an arithmetic sequence and predict the height in week 10.

• \(\text{Use the first and last heights: } u_1 = 12.0,\ u_5 = 20.0\)
• \(d = \dfrac{20.0 - 12.0}{4} = 2\)
• \(u_{10} = 12 + 9(2) = 30\)
• About 30 cm.

A swimmer does 20 lengths on day 1 and 4 more each day. (a) How many lengths on day 15? (b) How many lengths in total over the 15 days? Say which is a term and which is a sum.

• \(u_1 = 20,\ d = 4\)
• \((a)\ \text{A term: } u_{15} = 20 + 14(4) = 76\)
• \((b)\ \text{A sum: } S_{15} = \frac{15}{2}(20 + 76) = \frac{15}{2}(96) = 720\)
• 76 lengths on day 15; 720 lengths in total.

A pattern is made of squares. Stage 1 has 2 squares, stage 2 has 6 squares and stage 3 has 18 squares. (a) Is the number of squares arithmetic or geometric? Give the common difference or ratio. (b) How many squares are in stage 7? (c) How many squares are there in stages 1 to 7 altogether?

• \(2,\ 6,\ 18,\ \ldots\)
• (a) Each stage is 3 times the one before: geometric with \(u_1 = 2,\ r = 3\).
• \((b)\ u_{7} = 2 \times 3^{6} = 1458\)
• \((c)\ S_{7} = \dfrac{2(3^{7} - 1)}{3 - 1} = 2186\)