Edexcel PMT Mocks — Question 4 6 marks

Exam BoardEdexcel
ModulePMT Mocks (PMT Mocks)
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicArithmetic Sequences and Series
TypePeriodic or repeating sequence
DifficultyStandard +0.3 This question requires evaluating sin²(nπ/3) for specific values, recognizing the periodic pattern (period 6), then applying arithmetic series formula combined with the repeating sequence sum. While it involves multiple steps and pattern recognition, the techniques are standard A-level: exact trig values, sigma notation, and arithmetic series—no novel insight required.
Spec1.04g Sigma notation: for sums of series1.04h Arithmetic sequences: nth term and sum formulae1.05a Sine, cosine, tangent: definitions for all arguments
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 }\) is defined by
$$a _ { n } = \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right)$$ Find the exact values of
a. i) \(a _ { 1 }\) ii) \(a _ { 2 }\) iii) \(a _ { 3 }\) b. Hence find the exact value of $$\sum _ { n = 1 } ^ { 100 } \left\{ n + \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right) \right\}$$
(6 marks)
AnswerMarks Guidance
Part a.(3 marks)
B1\(a_1 = \frac{3}{4}\) Example: \(a_1 = \sin^2\left(\frac{\pi}{3}\right) = \frac{3}{4}\)
B1\(a_2 = \frac{3}{4}\) Example: \(a_2 = \sin^2\left(\frac{2\pi}{3}\right) = \frac{3}{4}\)
B1\(a_3 = 0\) Example: \(a_3 = \sin^2\left(\frac{3\pi}{3}\right) = 0\)
AnswerMarks
Part b.(3 marks)
M1Attempts a correct method to find the sum of \(1 + 2 + 3 + \cdots + 100\). Example: \(\frac{100}{2}(2 + 99)\) or \(\frac{100}{2}(1 + 100)\)
M1Attempts a correct method to find \(\sum_{n=1}^{100} \sin^2\left(\frac{n\pi}{3}\right)\). Example: \(66 \times \frac{3}{4} + \frac{3}{4}\) or \(33 \times \frac{3}{4} + 33 \times \frac{3}{4} + \frac{3}{4}\)
Must be a correct method for the correct sequence. Example: \(\frac{3}{4} + \frac{3}{4} + 0 + \frac{3}{4} + \frac{3}{4} + 0 + \frac{3}{4} + \frac{3}{4} + 0 + \cdots\)
AnswerMarks
A1\(5050 + \frac{201}{4} = \frac{20401}{4}\) or exact equivalent e.g. 5100.25 
(6 marks)

**Part a.** | (3 marks)

**B1** | $a_1 = \frac{3}{4}$ | Example: $a_1 = \sin^2\left(\frac{\pi}{3}\right) = \frac{3}{4}$

**B1** | $a_2 = \frac{3}{4}$ | Example: $a_2 = \sin^2\left(\frac{2\pi}{3}\right) = \frac{3}{4}$

**B1** | $a_3 = 0$ | Example: $a_3 = \sin^2\left(\frac{3\pi}{3}\right) = 0$

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**Part b.** | (3 marks)

**M1** | Attempts a correct method to find the sum of $1 + 2 + 3 + \cdots + 100$. Example: $\frac{100}{2}(2 + 99)$ or $\frac{100}{2}(1 + 100)$

**M1** | Attempts a correct method to find $\sum_{n=1}^{100} \sin^2\left(\frac{n\pi}{3}\right)$. Example: $66 \times \frac{3}{4} + \frac{3}{4}$ or $33 \times \frac{3}{4} + 33 \times \frac{3}{4} + \frac{3}{4}$

Must be a correct method for the correct sequence. Example: $\frac{3}{4} + \frac{3}{4} + 0 + \frac{3}{4} + \frac{3}{4} + 0 + \frac{3}{4} + \frac{3}{4} + 0 + \cdots$

**A1** | $5050 + \frac{201}{4} = \frac{20401}{4}$ or exact equivalent e.g. 5100.25

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\begin{enumerate}
  \item A sequence $a _ { 1 } , a _ { 2 } , a _ { 3 }$ is defined by
\end{enumerate}

$$a _ { n } = \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right)$$

Find the exact values of\\
a. i) $a _ { 1 }$\\
ii) $a _ { 2 }$\\
iii) $a _ { 3 }$\\
b. Hence find the exact value of

$$\sum _ { n = 1 } ^ { 100 } \left\{ n + \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right) \right\}$$

\hfill \mbox{\textit{Edexcel PMT Mocks  Q4 [6]}}
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