CAIE P1 2016 November — Question 5 6 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2016
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicGeometric Sequences and Series
TypeFind sum to infinity
DifficultyStandard +0.3 This is a straightforward geometric progression problem requiring students to set up two equations from given information (a + ar = 50, ar + ar² = 30), solve for the common ratio r, then apply the sum to infinity formula. While it involves multiple steps, the techniques are standard and the algebraic manipulation is routine for A-level students who have covered GP topics.
Spec1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1
5 The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30 . Find the sum to infinity.
Question 5:
AnswerMarks Guidance
\(a(1+r) = 50\) or \(\dfrac{a(1-r^2)}{1-r} = 50\)B1
\(ar(1+r) = 30\) or \(\dfrac{a(1-r^3)}{1-r} = 30 + a\)B1 Or otherwise attempt to solve for \(r\)
Eliminating \(a\) or \(r\)M1 Any correct method
\(r = 3/5\)A1
\(a = 125/4\) oeA1
\(S = 625/8\) oeA1\(\checkmark\) [6] Ft through on *their* \(r\) and \(a\) \((-1 < r < 1)\) 
## Question 5:
$a(1+r) = 50$ or $\dfrac{a(1-r^2)}{1-r} = 50$ | **B1** |
$ar(1+r) = 30$ or $\dfrac{a(1-r^3)}{1-r} = 30 + a$ | **B1** | Or otherwise attempt to solve for $r$
Eliminating $a$ or $r$ | **M1** | Any correct method
$r = 3/5$ | **A1** |
$a = 125/4$ oe | **A1** |
$S = 625/8$ oe | **A1**$\checkmark$ [6] | Ft through on *their* $r$ and $a$ $(-1 < r < 1)$

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5 The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30 . Find the sum to infinity.

\hfill \mbox{\textit{CAIE P1 2016 Q5 [6]}}
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