Sequences

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

1. The common ratio

   \(r = 2.4 \div 6 = 0.4\)

   Divide a term by the one before. Check: \(0.96 \div 2.4 = 0.4\).

2. The next two terms

   \(0.96 \times 0.4 = 0.384\), then \(0.384 \times 0.4 = 0.1536\)

   Multiply by 0.4 each time.

3. The general term

   \(u_n = 6 \times 0.4^{n-1}\)

   Use \(u_n = u_1 r^{n-1}\).

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

1. What the rule says

   \(u_{n+1} = 4u_n\)

   Each term is 4 times the term before.

2. Use it

   \(u_2 = 4u_1 = 4 \times 3 = 12\)

   Put \(u_1 = 3\) into the rule.

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

1. The rule

   Each term is the sum of the two before it.

2. 4th term

   \((f + 3g) + (2f + 3g) = 3f + 6g\)

   Add the 2nd and 3rd terms.

3. 5th term

   \((2f + 3g) + (3f + 6g) = 5f + 9g\)

   Add the 3rd and 4th terms.

4. 6th term

   \((3f + 6g) + (5f + 9g) = 8f + 15g\)

   Add the 4th and 5th terms.

Working to come.

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

1. The multiplier

   \(1 + \frac{2}{100} = 1.02\)

   Interest of 2% means the balance is multiplied by 1.02 each month.

2. The table

   \(1000 \times 1.02 = 1020\), \(1020 \times 1.02 = 1040.40\)

   Month 1: 1000. Month 2: 1020. Month 3: 1040.40. Month 4: 1061.21 to the nearest cent.

3. Show it is geometric

   \(1020 \div 1000 = 1.02\), \(1040.40 \div 1020 = 1.02\)

   Each balance divided by the one before is the same. A common ratio means the balances form a geometric sequence, with \(u_1 = 1000\) and \(r = 1.02\).

4. Why 1.02

   \(1 + 0.02 = 1.02\)

   The growth factor is 1 plus the interest rate written as a decimal. The 1 keeps what is already owed and the 0.02 adds the interest.

5. The general term

   \(u_n = 1000 \times 1.02^{n-1}\)

   The balance at the start of month \(n\). Check with month 3: \(1000 \times 1.02^2 = 1040.40\).

Working to come.