AQA Further Paper 2 2023 June — Question 7 3 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
Year2023
SessionJune
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSequences and series, recurrence and convergence
TypeFinding constants from given sum formula
DifficultyStandard +0.8 This requires knowing the standard cubic sum formula, manipulating the summation limits (subtracting the first 10 terms), and then factorizing a quartic into two quadratics with integer coefficients. While the steps are systematic, the algebraic manipulation and pattern recognition needed to express the result in the specific factored form elevates this above routine questions.
Spec4.06a Summation formulae: sum of r, r^2, r^3
Show that $$\sum_{r=11}^{n+1} r^3 = \frac{1}{4}(n^2 + an + b)(n^2 + an + c)$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
Question 7:
AnswerMarks
7Expresses the sum as
the difference of two
AnswerMarks Guidance
series.3.1a M1
∑ r3 = ∑ r3 −∑ r3
r=11 r=1 r=1
= 1(n+1 )2(n+2 )2 − 1( 10)2( 11 )2
4 4
= 1{(( n+1 )( n+2 )+110 )(( n+1 )( n+2 )−110 )}
4
= 1 {( n2 +3n+112 )( n2 +3n−108 )}
4
Obtains a correct
(unsimplified) expression
AnswerMarks Guidance
in terms of n for the sum.1.1b A1
Obtains the required
AnswerMarks Guidance
result.2.1 R1
Question total3
QMarking Instructions AO 
Question 7:
7 | Expresses the sum as
the difference of two
series. | 3.1a | M1 | n+1 n+1 10
∑ r3 = ∑ r3 −∑ r3
r=11 r=1 r=1
= 1(n+1 )2(n+2 )2 − 1( 10)2( 11 )2
4 4
= 1{(( n+1 )( n+2 )+110 )(( n+1 )( n+2 )−110 )}
4
= 1 {( n2 +3n+112 )( n2 +3n−108 )}
4
Obtains a correct
(unsimplified) expression
in terms of n for the sum. | 1.1b | A1
Obtains the required
result. | 2.1 | R1
Question total | 3
Q | Marking Instructions | AO | Marks | Typical solution
Show that
$$\sum_{r=11}^{n+1} r^3 = \frac{1}{4}(n^2 + an + b)(n^2 + an + c)$$

where $a$, $b$ and $c$ are integers to be found.
[3 marks]

\hfill \mbox{\textit{AQA Further Paper 2 2023 Q7 [3]}}
This paper (16 questions)
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Q1 1 Q2 1 Q3 1 Q4 1 Q5 5 Q6 5 Q7 3 Q8 6 Q9 7 Q10 8 Q11 9 Q12 6 Q13 11 Q14 10 Q15 10 Q16 16