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 πth term 5π+2 . 2π Show that the limiting value of the sequence S, as π → ∞ is 2.5 3. [Set 1 Paper 2 Q3] (a) The πth term of a sequence is 4π − 10. (a)(i) Show that the (π + 1)th term can be written as 4π − 6. (a)(ii) Prove that the sum of any two consecutive terms of the sequence is divisible by 8. (b) The πth term of a different 3π sequence is π+5 (b)(i) Explain why 1 is not a term in this sequence. (b)(ii) Work out the limiting value of the sequence as π → ∞. 4. [Set 2 Paper 1 Q2] Here is a linear sequence: 4 11 18 25 … (a) Work out an expression for the πth term. (b) How many terms are less than 150? 5. [Set 2 Paper 2 Q11] The πth term of sequence X is ππ + π. The πth term of sequence Y is ππ + π. (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 π = 2π. (c) Prove that: ππ‘β π‘πππ ππ π πππ’ππππ π 2π + 1 = ππ‘β π‘πππ ππ π πππ’ππππ π π+2 6. [Set 4 Paper 2 Q11] (a) π(π) = π2 + π Show that π(π + 1) − π(π) = 2π + 2 (b) The πth term of a sequence is π2 + π Two consecutive terms in the sequence have a difference of 32. Work out the two terms. 7. [June 2012 Paper 1 Q10] The πth term of the linear sequence 2, 7, 12, 17, … is 5π − 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 πth term of the linear sequence 4 7 10 13 … (b) Hence, write down the πth 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 πth term. π2 + 5π + 6 10. A linear sequence starts π + π π + 3π π + 5π π + 7π The 5th and 8th terms have values 35 and 59. (a) Work out π and π. (b) Work out the πth term of the sequence. 11. [AQA Worksheet] A sequence has nth term 3π+1 . π (a) Show that the difference between the πth and (π + 1)th term is 1 π(π+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 π → ∞.