Arithmetic sequences Problem set B Worked solutions

Find the term asked for.

1. (a) \(u_1 = 11,\ d = 7\). Find \(u_{14}\).

   • \(u_n = u_1 + (n-1)d\)
   • \(u_{14} = 11 + 13(7) = 102\)

2. (b) 70, 63, 56, … Find \(u_{18}\).

   • \(d = 63 - 70 = -7\)
   • \(u_{18} = 70 + 17(-7) = -49\)

3. (c) 1.5, 4, 6.5, … Find \(u_{27}\).

   • \(d = 4 - 1.5 = 2.5\)
   • \(u_{27} = 1.5 + 26(2.5) = 66.5\)

The 5th term of an arithmetic sequence is 19 and the 11th term is 43. Find the first term and the common difference.

• \(u_{11} - u_5 = 6d\)
• \(43 - 19 = 6d,\ d = 4\)
• \(u_1 = u_5 - 4d = 19 - 16 = 3\)

Write an expression for \(u_n\).

1. (a) 9, 15, 21, …

   • \(u_n = u_1 + (n-1)d\)
   • \(u_1 = 9,\ d = 6\)
   • \(u_n = 9 + 6(n-1) = 6n + 3\)

2. (b) 50, 45, 40, …

   • \(u_n = u_1 + (n-1)d\)
   • \(u_1 = 50,\ d = -5\)
   • \(u_n = 50 + (-5)(n-1) = 55 - 5n\)

3. (c) −9, −5, −1, …

   • \(u_n = u_1 + (n-1)d\)
   • \(u_1 = -9,\ d = 4\)
   • \(u_n = -9 + 4(n-1) = 4n - 13\)

Answer each part.

1. (a) Which term of 3, 11, 19, … is equal to 235?

   • \(3 + 8(n-1) = 235\)
   • \(8(n-1) = 232,\ n - 1 = 29\)
   • \(n = 30\)

2. (b) Find the first negative term of 92, 87, 82, …

   • \(92 - 5(n-1) < 0\)
   • \(n - 1 > 18.4,\ \text{so } n = 20\)
   • \(u_{20} = 92 - 5(19) = -3\)

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

1. (a) \(u_3 = 37,\ u_9 = 7\)

   • \(6d = u_9 - u_3 = 7 - 37 = -30\)
   • \(d = -5\)
   • \(u_1 = u_3 - 2d = 37 + 10 = 47\)

2. (b) \(u_7 = 16,\ u_{17} = 56\); find \(u_{32}\)

   • \(10d = 56 - 16 = 40\)
   • \(d = 4,\ u_1 = 16 - 6(4) = -8\)
   • \(u_{32} = -8 + 31(4) = 116\)

Say which of these numbers are terms of the sequence 8, 14, 20, 26, …, and give the position of each one that is.

1. (a) 302

   • \(8 + (n - 1)(6) = 302\)
   • \(n = 50\)
   • Whole number: 302 is term 50.

2. (b) 305

   • \(8 + (n - 1)(6) = 305\)
   • \(n = 50.50\ldots\)
   • Not a whole number: 305 is not a term.

3. (c) 164

   • \(8 + (n - 1)(6) = 164\)
   • \(n = 27\)
   • Whole number: 164 is term 27.

A garden is planted in rows. The first row has 14 plants and each row has 2 more plants than the row before. (a) How many plants are in row 18? (b) Which row has 60 plants?

• \(u_n = 14 + 2(n-1)\)
• \(u_{18} = 14 + 17(2) = 48\)
• \(14 + 2(n-1) = 60,\ n - 1 = 23\)
• (a) 48 plants (b) row 24

A water tank holds 80 litres and loses 2.5 litres every minute. (a) Write an expression for the volume after \(n\) minutes. (b) How long until it is empty?

• \(u_n = 80 - 2.5n\)
• \(80 - 2.5n = 0\)
• \(n = 32\)
• (a) \(80 - 2.5n\) litres (b) 32 minutes

Tiles are laid in a row and edged with strips. One tile needs 6 strips, two tiles need 10 strips and three tiles need 14 strips. (a) How many strips are needed to edge a row of 18 tiles? (b) A shop has 155 strips. What is the greatest number of tiles it can edge in one row?

• \(6,\ 10,\ 14,\ \ldots \quad u_1 = 6,\ d = 4\)
• \(u_n = 6 + (n - 1)(4) = 4n + 2\)
• \((a)\ u_{18} = 4(18) + 2 = 74\)
• \((b)\ 4n + 2 \leq 155 \Rightarrow n \leq 38.25\)
• The greatest number is 38 tiles (it uses 154 strips).

\(x,\ 3x + 1\) and \(6x - 5\) are three consecutive terms of an arithmetic sequence. Find \(x\) and the three terms.

• \((3x + 1) - x = (6x - 5) - (3x + 1)\)
• \(2x + 1 = 3x - 6\)
• \(x = 7\)
• The terms are 7, 22, 37.

How many multiples of 9 are there between 200 and 600?

• The first is 207 and the last is 594.
• \(207 + 9(n-1) = 594\)
• \(n - 1 = 43\)
• There are 44 multiples.