OCR FP1 2012 June — Question 4 7 marks

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2012
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSequences and series, recurrence and convergence
TypeStandard summation formulae application
DifficultyModerate -0.8 This is a straightforward application of standard summation formulae (∑r², ∑r, ∑1) with basic algebraic manipulation and factorisation. While it's Further Maths content, it requires only direct substitution of known formulae and routine algebraic simplification, making it easier than average even for FP1 students.
Spec4.06a Summation formulae: sum of r, r^2, r^3

4 Find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 2 \right)\), expressing your answer in a fully factorised form.

Question 4:
AnswerMarks Guidance
M1Express as sum of 3 series
M1Use standard series results, at least 1 correct
A1Two terms correct
\(\frac{1}{2}n(n+1)(2n+1) - \frac{3}{2}n(n+1) + 2n\)A1 Third term correct
M1Obtain factor of \(n\)
\(n(n^2 + 1)\)A2 Obtain correct answer c.a.o. (Allow A1 for \(\frac{1}{2(2n^2+2)}\))
[7]
## Question 4:

| M1 | Express as sum of 3 series
| M1 | Use standard series results, at least 1 correct
| A1 | Two terms correct
$\frac{1}{2}n(n+1)(2n+1) - \frac{3}{2}n(n+1) + 2n$ | A1 | Third term correct
| M1 | Obtain factor of $n$
$n(n^2 + 1)$ | A2 | Obtain correct answer c.a.o. (Allow A1 for $\frac{1}{2(2n^2+2)}$)
**[7]**

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4 Find $\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } - 3 r + 2 \right)$, expressing your answer in a fully factorised form.

\hfill \mbox{\textit{OCR FP1 2012 Q4 [7]}}