CAIE P1 2018 June — Question 3 5 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2018
SessionJune
Marks5
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Mark schemeDownload PDF ↗
TopicGeometric Sequences and Series
TypeFind N for S_∞ - S_N condition
DifficultyStandard +0.3 This is a straightforward application of standard GP formulas (S_n and S_∞) with a given common ratio. The only steps are substituting r=0.99 and n=100 into the formulas, forming a ratio, and converting to a percentage. It requires routine recall and basic calculation, making it slightly easier than average since there's no problem-solving or conceptual challenge involved.
Spec1.04i Geometric sequences: nth term and finite series sum1.04j Sum to infinity: convergent geometric series |r|<1
3 The common ratio of a geometric progression is 0.99 . Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.
Question 3:
AnswerMarks Guidance
\(\left[\frac{a(1-r^n)}{1-r}\right] \div \left[\frac{a}{1-r}\right]\)M1M1 Correct formulae used with/without \(r = 0.99\) or \(n = 100\)
(division of \(S_n\) by \(S_\infty\))DM1 Allow numerical \(a\) (M1M1). 3rd M1 is for division \(\frac{S_n}{S_\infty}\) (or ratio) SOI
\(1 - 0.99^{100}\) SOI OR \(\frac{63(a)}{100(a)}\) SOIA1 Could be shown multiplied by 100(%). Dep. on DM1
63(%) Allow 63.4 or 0.63 but not 2 infringements (e.g. 0.634, 0.63%)A1 \(n = 99\) used scores Max M3. Condone \(a = 0.99\) throughout. \(S_n = S_\infty\) (without division shown) scores 2/5
Total: 5 
## Question 3:

| $\left[\frac{a(1-r^n)}{1-r}\right] \div \left[\frac{a}{1-r}\right]$ | M1M1 | Correct formulae **used** with/without $r = 0.99$ or $n = 100$ |
|---|---|---|
| (division of $S_n$ by $S_\infty$) | DM1 | Allow numerical $a$ (M1M1). 3rd M1 is for division $\frac{S_n}{S_\infty}$ (or ratio) SOI |
| $1 - 0.99^{100}$ SOI OR $\frac{63(a)}{100(a)}$ SOI | A1 | Could be shown multiplied by 100(%). Dep. on DM1 |
| 63(%) Allow 63.4 or 0.63 but not 2 infringements (e.g. 0.634, 0.63%) | A1 | $n = 99$ used scores Max M3. Condone $a = 0.99$ throughout. $S_n = S_\infty$ (without division shown) scores 2/5 |
| **Total: 5** | | |

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3 The common ratio of a geometric progression is 0.99 . Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.\\

\hfill \mbox{\textit{CAIE P1 2018 Q3 [5]}}
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