Sequences Worked Examples for IGCSE Maths

Question

Find the nth term of the sequence 7, 11, 15, 19, 23, ...

Solution

1. 1

   Find the common difference

   11-7 = 4, 15-11 = 4. Common difference d = 4.

   Subtract consecutive terms to find the common difference.

2. 2

   Write the nth term

   nth term = 4n + (7-4) = 4n + 3

   nth term = dn + (first term - d).

3. 3

   Verify

   n=1: 4(1)+3 = 7. n=2: 4(2)+3 = 11. n=3: 4(3)+3 = 15. All correct.

   Always check your formula against the given terms.

Question

Find the nth term of the sequence 3, 9, 19, 33, 51, ...

Solution

1. 1

   Find first and second differences

   First differences: 6, 10, 14, 18. Second differences: 4, 4, 4.

   Constant second difference means it is a quadratic sequence.

2. 2

   Find the coefficient of n squared

   Coefficient of n squared = second difference / 2 = 4/2 = 2. So the sequence starts with 2n squared.

   For a quadratic sequence an squared + bn + c, a = second difference / 2.

3. 3

   Subtract 2n squared from each term

   n=1: 3-2=1, n=2: 9-8=1, n=3: 19-18=1, n=4: 33-32=1, n=5: 51-50=1

   The remainders form a constant sequence.

4. 4

   Write the nth term

   nth term = 2n squared + 1

   Since the remainders are all 1, the linear part is 0n + 1.

Question

Find the sum of the first 20 terms of the sequence 5, 8, 11, 14, ...

Solution

1. 1

   Identify a and d

   First term a = 5, common difference d = 3

   These are needed for the sum formula.

2. 2

   Use the sum formula

   S(n) = n/2 (2a + (n-1)d) = 20/2 (2(5) + 19(3)) = 10(10 + 57) = 10(67) = 670

   Substitute n = 20, a = 5, d = 3 into the formula.