IGCSE Further Maths Sequences
Exercise 1
1. [Worksheet Q1] A linear sequence
starts
250 246 242 238
Which term is the first to have a
negative value?
2. [AQA Worksheet] A sequence has 𝑛th
term
5𝑛+2
.
2𝑛
Show that the limiting
value of the sequence S, as 𝑛 → ∞ is
2.5
3. [Set 1 Paper 2 Q3]
(a) The 𝑛th term of a sequence is
4𝑛 − 10.
(a)(i) Show that the (𝑛 + 1)th term
can be written as 4𝑛 − 6.
(a)(ii) Prove that the sum of any two
consecutive terms of the sequence is
divisible by 8.
(b) The 𝑛th term of a different
3𝑛
sequence is 𝑛+5
(b)(i) Explain why 1 is not a term in
this sequence.
(b)(ii) Work out the limiting value of
the sequence as 𝑛 → ∞.
4. [Set 2 Paper 1 Q2] Here is a linear
sequence:
4 11 18 25 …
(a) Work out an expression for the
𝑛th term.
(b) How many terms are less than
150?
5. [Set 2 Paper 2 Q11]
The 𝑛th term of sequence X is 𝑎𝑛 + 𝑏.
The 𝑛th term of sequence Y is 𝑏𝑛 + 𝑎.
(a) Show that the sequences have the
same first term.
(b) The 2nd term of sequence X is
equal to the 3rd term of sequence Y.
Show that 𝑎 = 2𝑏.
(c) Prove that:
𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑋 2𝑛 + 1
=
𝑛𝑡ℎ 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑌
𝑛+2
6. [Set 4 Paper 2 Q11]
(a) 𝑓(𝑛) = 𝑛2 + 𝑛 Show that
𝑓(𝑛 + 1) − 𝑓(𝑛) = 2𝑛 + 2
(b) The 𝑛th term of a sequence is
𝑛2 + 𝑛
Two consecutive terms in the
sequence have a difference of 32.
Work out the two terms.
7. [June 2012 Paper 1 Q10] The 𝑛th
term of the linear sequence 2, 7, 12,
17, … is 5𝑛 − 3. A new sequence is
formed by squaring each term of the
linear sequence and adding 1. Prove
algebraically that all the terms in the
new sequence are multiples of 5.
8. [Worksheet Q4]
(a) Write down the 𝑛th term of the
linear sequence
4 7 10 13 …
(b) Hence, write down the 𝑛th term of
the quadratic sequence.
16 49 100 169 …
(c) For the sequence in (b), show that
the 30th term is equal to the product
of the 2nd and 4th.
9. [AQA Worksheet]
This pattern of rectangles continues.
Show that the sequence of numbers
formed by the areas of these
rectangles has 𝑛th term.
𝑛2 + 5𝑛 + 6
10. A linear sequence starts
𝑎 + 𝑏 𝑎 + 3𝑏 𝑎 + 5𝑏 𝑎 + 7𝑏
The 5th and 8th terms have values 35
and 59.
(a) Work out 𝑎 and 𝑏.
(b) Work out the 𝑛th term of the
sequence.
11. [AQA Worksheet] A sequence has nth
term
3𝑛+1
.
𝑛
(a) Show that the difference between
the 𝑛th and (𝑛 + 1)th term is
1
𝑛(𝑛+1)
(b) Which are the first two
consecutive terms with a difference
less than 0.01?
(c) Write down the limiting value of
the sequence as 𝑛 → ∞.