Arithmetic sequences Problem set A Worked solutions

Find the term asked for.

1. (a) \(u_1 = 9,\ d = 6\). Find \(u_{15}\).

   • \(u_{15} = u_1 + (15 - 1)d = 9 + 14(6)\)
   • \(u_{15} = 93\)

2. (b) 52, 45, 38, … Find \(u_{20}\).

   • \(d = 45 - 52 = -7\)
   • \(u_{20} = 52 + 19(-7)\)
   • \(u_{20} = -81\)

3. (c) 2.5, 4, 5.5, … Find \(u_{31}\).

   • \(d = 4 - 2.5 = 1.5\)
   • \(u_{31} = 2.5 + 30(1.5)\)
   • \(u_{31} = 47.5\)

The 4th term of an arithmetic sequence is 20 and the 9th term is 45. Find the first term and the common difference.

• \(u_4 = u_1 + 3d = 20 \quad\text{and}\quad u_9 = u_1 + 8d = 45\)
• \(\text{Subtract: } 5d = 25 \Rightarrow d = 5\)
• \(u_1 = 20 - 3(5) = 5\)

Write an expression for \(u_n\).

1. (a) 8, 13, 18, …

   • \(u_1 = 8,\ d = 5\)
   • \(u_n = 8 + (n - 1)(5)\)
   • \(u_n = 5n + 3\)

2. (b) 30, 26, 22, …

   • \(u_1 = 30,\ d = -4\)
   • \(u_n = 30 + (n - 1)(-4)\)
   • \(u_n = 34 - 4n\)

3. (c) −7, −4, −1, …

   • \(u_1 = -7,\ d = 3\)
   • \(u_n = -7 + (n - 1)(3)\)
   • \(u_n = 3n - 10\)

Answer each part.

1. (a) Which term of 5, 12, 19, … is equal to 187?

   • \(d = 7,\ u_n = 5 + (n - 1)(7) = 187\)
   • \((n - 1) = 182/7 = 26\)
   • 187 is term number 27.

2. (b) Find the first negative term of 80, 77, 74, …

   • \(d = -3,\ u_n = 80 + (n - 1)(-3) = 83 - 3n\)
   • \(83 - 3n < 0 \Rightarrow n > \frac{83}{3} = 27.67\ldots\)
   • \(n = 28: u_{28} = 83 - 84 = -1\)
   • The 28th term, −1, is the first negative term.

Find \(u_1\) and \(d\), and then the term asked for.

1. (a) \(u_3 = 31,\ u_{11} = -1\)

   • \(u_{11} - u_3 = 8d = -32\)
   • \(d = -4\)
   • \(u_1 = u_3 - 2d = 31 - 2(-4) = 39\)

2. (b) \(u_6 = 10,\ u_{15} = 37\); find \(u_{30}\)

   • \(u_{15} - u_6 = 9d = 27\)
   • \(d = 3\)
   • \(u_1 = u_6 - 5d = 10 - 5(3) = -5\)
   • \(u_{30} = -5 + 29(3) = 82\)

Say which of these numbers are terms of the sequence 5, 12, 19, 26, …, and give the position of each one that is.

1. (a) 306

   • \(5 + (n - 1)(7) = 306\)
   • \(n = 44\)
   • Whole number: 306 is term 44.

2. (b) 310

   • \(5 + (n - 1)(7) = 310\)
   • \(n = 44.57\ldots\)
   • Not a whole number: 310 is not a term.

3. (c) 152

   • \(5 + (n - 1)(7) = 152\)
   • \(n = 22\)
   • Whole number: 152 is term 22.

A theatre has 18 seats in its front row. Each row has 3 more seats than the row in front. (a) How many seats are in row 14? (b) Which row has 90 seats?

• \(u_1 = 18,\ d = 3\)
• \((a)\ u_{14} = 18 + 13(3) = 57\)
• \((b)\ 18 + (n - 1)(3) = 90 \Rightarrow n - 1 = 24 \Rightarrow n = 25\)
• (a) 57 seats. (b) Row 25.

A candle is 24 cm tall and burns down 1.5 cm every hour. (a) Write an expression for its height after \(n\) hours. (b) How long does it last?

• \(\text{Start } 24,\ d = -1.5\)
• \((a)\ h_n = 24 - 1.5n \text{ cm}\)
• \((b)\ 24 - 1.5n = 0 \Rightarrow n = \frac{24}{1.5} = 16\)
• It lasts 16 hours.

Square tables are pushed together in a row. One table seats 4 people, two tables seat 6 and three tables seat 8. (a) How many people can sit at a row of 15 tables? (b) A club needs seats for 91 people. How many tables must be in the row?

• \(4,\ 6,\ 8,\ \ldots \quad u_1 = 4,\ d = 2\)
• \(u_n = 4 + (n - 1)(2) = 2n + 2\)
• \((a)\ u_{15} = 2(15) + 2 = 32\)
• \((b)\ 2n + 2 \geq 91 \Rightarrow n \geq 44.5\)
• The row needs 45 tables (44 tables seat only 90).

\(x,\ 2x + 3\) and \(5x - 4\) are three consecutive terms of an arithmetic sequence. Find \(x\) and the three terms.

• \((2x + 3) - x = (5x - 4) - (2x + 3)\)
• \(x + 3 = 3x - 7\)
• \(2x = 10 \Rightarrow x = 5\)
• Terms: 5, 13, 21 (d = 8).

How many multiples of 7 are there between 100 and 500?

• First multiple of 7 above 100: 105. Last below 500: 497.
• \(105 + (n - 1)(7) = 497\)
• \(n - 1 = 56 \Rightarrow n = 57\)
• 57 multiples.