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 → ∞.