Sequences

Have a go first. Then open the working one line at a time.

1. The first term

   \(u_1 = 7\)

   Read it from the sequence.

2. The common difference

   \(d = 5\)

   Each term is 5 more than the one before.

3. The general term

   \(u_n = u_1 + (n-1)d\) \(= 7 + (n-1)5\) \(= 5n + 2\)

   Put both into the general term.

4. Check

   \(5(4) + 2 = 22\)

   \(n = 4\) should give the fourth term, 22.

Have a go first. Then open the working one line at a time.

1. Show it is arithmetic

   \(12 - 5 = 7\), \(19 - 12 = 7\), \(26 - 19 = 7\)

   The difference is always 7, so the sequence is arithmetic with \(d = 7\).

2. The general term

   \(u_n = u_1 + (n-1)d\) \(= 5 + (n-1)7\) \(= 7n - 2\)

   Use \(u_1 = 5\) and \(d = 7\).

3. The 100th term

   \(u_{100} = 7(100) - 2 = 698\)

   Use the rule with \(n = 100\).

4. Is 355 a term?

   \(7n - 2 = 355\) gives \(n = 51\)

   A whole number, so 355 is the 51st term.

5. Is 1000 a term?

   \(7n - 2 = 1000\) gives \(n = 143.14\ldots\)

   Not a whole number, so 1000 is not a term.

Have a go first. Then open the working one line at a time.

1. Show it is arithmetic

   \(81 - 87 = -6\), \(75 - 81 = -6\), \(69 - 75 = -6\)

   The difference is always −6, so the sequence is arithmetic with \(d = -6\).

2. The general term

   \(u_n = 87 + (n-1)(-6)\) \(= 93 - 6n\)

   Use \(u_1 = 87\) and \(d = -6\).

3. The 100th term

   \(u_{100} = 93 - 600 = -507\)

   Use the rule with \(n = 100\).

4. Is −99 a term?

   \(93 - 6n = -99\) gives \(n = 32\)

   A whole number, so −99 is the 32nd term.

5. Is 50 a term?

   \(93 - 6n = 50\) gives \(n = 7.16\ldots\)

   Not a whole number, so 50 is not a term.

Have a go first. Then open the working one line at a time.

1. 1, __, 15, __, 29

   1, 8, 15, 22, 29

   From 1 to 15 is 2 steps and a gap of 14, so \(d = 7\).

2. −19, __, __, 68, __

   −19, 10, 39, 68, 97

   From −19 to 68 is 3 steps and a gap of 87, so \(d = 29\).

Have a go first. Then open the working one line at a time.

1. Equal differences

   \(5 - (k + 8) = (3k - 2) - 5\)

   In an arithmetic sequence the two differences are equal.

2. Solve

   \(-3 - k = 3k - 7\), so \(4k = 4\) and \(k = 1\)

   Simplify each side, then collect the \(k\) terms.

3. Check

   9, 5, 1

   The terms are 9, 5, 1, with difference −4 each time.

Have a go first. Then open the working one line at a time.

1. Find the common difference

   \(d = 20 \div 5 = 4\)

   From the 20th term to the 25th is 5 steps, and the terms go from 75 to 95, a gap of 20.

2. Find the 48th term

   \(u_{48} = 95 + 23 \times 4 = 187\)

   From the 25th term to the 48th is 23 steps.