The answer
(a)(i) \(6n + 5\)
(ii) remainder 2 on division by 3
(b)(i) \(\dfrac{19}{180}\)
(ii) \(k = 9\)
(iii) \(n = 23\)
O-Level E-Math 2018 Paper 2 Question 4 · Verified worked solution by the Genius Plus Academy teaching team
The question
(a) The first four terms are \(11, 17, 23, 29\). (i) Find an expression, in terms of \(n\), for the \(n\)th term. [2] (ii) Explain why no term can be a multiple of 3. [1]
(b) A different sequence has \(T_n = \dfrac{4n - 1}{205 - 5n}\). (i) Find \(T_5\) as a fraction. [1] (ii) \(T_k = \dfrac{7}{32}\). Find \(k\). [3] (iii) Find the least \(n\) for which \(T_n > 1\). [3]
(a)(i) The common difference is \(6\), and the term before the first would be \(11 - 6 = 5\), so the \(n\)th term is \(6n + 5\). (Check: \(n = 1\) gives 11, \(n = 4\) gives 29.)
(ii) \(6n + 5 = 3(2n + 1) + 2\). Dividing by 3 always leaves a remainder of 2, so the term is never exactly divisible by 3, i.e. never a multiple of 3.
(b)(i) \(T_5 = \dfrac{4(5) - 1}{205 - 5(5)} = \dfrac{19}{180}\).
(ii) \(\dfrac{4k - 1}{205 - 5k} = \dfrac{7}{32}\). Cross-multiply: \(32(4k - 1) = 7(205 - 5k) \Rightarrow 128k - 32 = 1435 - 35k \Rightarrow 163k = 1467 \Rightarrow k = 9\). (Check: \(T_9 = \dfrac{35}{160} = \dfrac{7}{32}\).)
(iii) For \(n < 41\) the denominator \(205 - 5n > 0\), so \(T_n > 1\) requires \(4n - 1 > 205 - 5n \Rightarrow 9n > 206 \Rightarrow n > 22.9\). The least integer is \(n = 23\). (Check: \(T_{22} = \dfrac{87}{95} < 1\), \(T_{23} = \dfrac{91}{90} > 1\).)
Answer: (a)(i) \(6n + 5\)
(ii) remainder 2 on division by 3
(b)(i) \(\dfrac{19}{180}\)
(ii) \(k = 9\)
(iii) \(n = 23\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
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It is a linear (nth-term) rule question from Sequences, worth 10 marks: (a) 2 + 1, (b) 1 + 3 + 3.
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