The answer
(a) \(T_5 = 53\)
(b) odd \(+\) even structure (shown)
(c) \(T_n = n^2 + 5n + 3\) (shown)
(d) \(2p + 6\)
(e) \(2p + 6 = 4\) has no valid \(p\)
O-Level E-Math 2016 Paper 2 Question 4 · Verified worked solution by the Genius Plus Academy teaching team
What this question tests
This is Question 4 of the O-Level E-Math 2016 Paper 2. It tests continue a pattern, in the Sequences area. It is worth 9 marks: 1 + 1 + 3 + 3 + 1. It is a worded / diagram-based question, so open your Ten-Year Series (TYS) or the official paper at this question, then follow our full worked solution below.
(a) Following the pattern, the base is squared and a number is added; \(T_5 = 6^2 + 17 = 36 + 17 = 53\). (The added numbers \(5, 8, 11, 14, 17\) go up by 3 each time.)
(b) Write the general term as \(T_n = (n + 1)^2 + (3n + 2)\) (the square base is \(n + 1\), and the added number is \(5 + 3(n - 1) = 3n + 2\)). Now \((n + 1)^2\) and \(3n + 2\) always have opposite parity: if \(n\) is odd then \((n+1)^2\) is even and \(3n + 2\) is odd; if \(n\) is even then \((n+1)^2\) is odd and \(3n + 2\) is even. An even plus an odd is always odd, so \(T_n\) is always odd.
(c) \(T_n = (n + 1)^2 + (3n + 2) = (n^2 + 2n + 1) + (3n + 2) = n^2 + 5n + 3\). (shown) (Check: \(n = 1\) gives \(1 + 5 + 3 = 9\); \(n = 4\) gives \(16 + 20 + 3 = 39\), matching.)
(d) \(T_{p+1} - T_p = \big[(p + 1)^2 + 5(p + 1) + 3\big] - \big[p^2 + 5p + 3\big]\). Expand the first bracket: \(p^2 + 2p + 1 + 5p + 5 + 3 = p^2 + 7p + 9\). Subtract: \((p^2 + 7p + 9) - (p^2 + 5p + 3) = 2p + 6\).
(e) A difference of 4 would need \(2p + 6 = 4\), i.e. \(p = -1\), which is not a valid term position (\(p\) must be a positive integer). The smallest actual difference is \(T_2 - T_1 = 8\), and the differences only increase, so no two consecutive terms differ by 4.
Answer: (a) \(T_5 = 53\)
(b) odd \(+\) even structure (shown)
(c) \(T_n = n^2 + 5n + 3\) (shown)
(d) \(2p + 6\)
(e) \(2p + 6 = 4\) has no valid \(p\)
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
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a continue a pattern question from Sequences, worth 9 marks: 1 + 1 + 3 + 3 + 1.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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