OCR FP1 2011 January — Question 4 6 marks

Exam BoardOCR
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJanuary
Marks6
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Mark schemeDownload PDF ↗
TopicSequences and Series
TypeFinding Constants from Identity
DifficultyStandard +0.8 This is a Further Maths FP1 question requiring students to equate coefficients by expanding both the summation formulas (using standard results for Σr³ and Σr) and the RHS polynomial, then solve simultaneously. While it uses standard summation formulas, the algebraic manipulation and systematic coefficient comparison across a quartic polynomial requires careful work beyond typical A-level, placing it moderately above average difficulty.
Spec4.06a Summation formulae: sum of r, r^2, r^3
4 Given that \(\sum _ { r = 1 } ^ { n } \left( a r ^ { 3 } + b r \right) \equiv n ( n - 1 ) ( n + 1 ) ( n + 2 )\), find the values of the constants \(a\) and \(b\).
Question 4:
AnswerMarks Guidance
AnswerMarks Guidance
B1Correct value for \(\sum r\) stated or used
M1Express as sum of two series
\(\frac{a}{4}n^2(n+1)^2 + \frac{bn}{2}(n+1)\)A1 Obtain correct unsimplified answer
M1Compare coefficients or substitute values for \(n\)
\(a = 4, \quad b = -4\)A1 A1 6 Obtain correct answers
*Or*
\(a + b = 0, \quad 4a + b = 12\)M1, A1 A1 Use 2 values for \(n\); obtain correct equations
\(a = 4, \quad b = -4\)M1, A1 A1 Solve simultaneous equations; obtain correct answers 
## Question 4:

| Answer | Marks | Guidance |
|--------|-------|----------|
| | B1 | Correct value for $\sum r$ stated or used |
| | M1 | Express as sum of two series |
| $\frac{a}{4}n^2(n+1)^2 + \frac{bn}{2}(n+1)$ | A1 | Obtain correct unsimplified answer |
| | M1 | Compare coefficients or substitute values for $n$ |
| $a = 4, \quad b = -4$ | A1 A1 **6** | Obtain correct answers |
| *Or* | | |
| $a + b = 0, \quad 4a + b = 12$ | M1, A1 A1 | Use 2 values for $n$; obtain correct equations |
| $a = 4, \quad b = -4$ | M1, A1 A1 | Solve simultaneous equations; obtain correct answers |

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4 Given that $\sum _ { r = 1 } ^ { n } \left( a r ^ { 3 } + b r \right) \equiv n ( n - 1 ) ( n + 1 ) ( n + 2 )$, find the values of the constants $a$ and $b$.

\hfill \mbox{\textit{OCR FP1 2011 Q4 [6]}}
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Q1 7 Q2 6 Q3 4 Q4 6 Q5 3 Q6 8 Q7 9 Q8 9 Q9 9 Q10 11