Edexcel C2 2005 June — Question 9 10 marks

Exam BoardEdexcel
ModuleC2 (Core Mathematics 2)
Year2005
SessionJune
Marks10
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Mark schemeDownload PDF ↗
TopicGeometric Sequences and Series
TypeProve sum formula
DifficultyEasy -1.2 This is a straightforward C2 question testing standard geometric series knowledge. Part (a) is a bookwork proof that should be memorized. Parts (b) and (c) are direct applications of the GP formula with simple arithmetic—finding the 4th term and sum of 20 terms respectively. No problem-solving insight required, just routine application of formulas.
Spec1.04i Geometric sequences: nth term and finite series sum
9. (a) A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ Mr. King will be paid a salary of \(\pounds 35000\) in the year 2005 . Mr. King's contract promises a \(4 \%\) increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
(b) Find, to the nearest \(\pounds 100\), Mr. King's salary in the year 2008. Mr. King will receive a salary each year from 2005 until he retires at the end of 2024.
(c) Find, to the nearest \(\pounds 1000\), the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024.
Question 9:
(a)
AnswerMarks Guidance
AnswerMarks Guidance
\(S = a + ar + \ldots + ar^{n-1}\)B1 At least 3 terms shown, no extra terms; "\(S=\)" not required
\(rS = ar + ar^2 + \ldots + ar^n\)M1 "\(rS=\)" not required; multiply by \(r\)
\(S(1-r) = a(1-r^n)\), \(S = \frac{a(1-r^n)}{1-r}\)M1, A1cso Subtract and factorise; completely correct solution; (4 marks)
(b)
AnswerMarks Guidance
AnswerMarks Guidance
\(ar^{n-1} = 35000 \times 1.04^3 = 39400\)M1, A1 Correct \(a\) and \(r\), with \(n = 3, 4\) or \(5\); (2 marks) Answer only: 39400 full marks, 39370 scores M1 A0
(c)
AnswerMarks Guidance
AnswerMarks Guidance
\(n = 20\)B1 Seen or implied
\(S_{20} = \frac{35000(1-1.04^{20})}{(1-1.04)}\)M1, A1ft Needs any \(r\), \(a=35000\), \(n=19\), 20 or 21; A1ft from \(n=19\) or \(n=21\) but \(r\) must be 1.04
\(= 1\,042\,000\)A1 (4 marks)
*Note: Failure to round correctly in (b) and (c): penalise once only (first occurrence).*
## Question 9:

**(a)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $S = a + ar + \ldots + ar^{n-1}$ | B1 | At least 3 terms shown, no extra terms; "$S=$" not required |
| $rS = ar + ar^2 + \ldots + ar^n$ | M1 | "$rS=$" not required; multiply by $r$ |
| $S(1-r) = a(1-r^n)$, $S = \frac{a(1-r^n)}{1-r}$ | M1, A1cso | Subtract and factorise; completely correct solution; (4 marks) |

**(b)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $ar^{n-1} = 35000 \times 1.04^3 = 39400$ | M1, A1 | Correct $a$ and $r$, with $n = 3, 4$ or $5$; (2 marks) Answer only: 39400 full marks, 39370 scores M1 A0 |

**(c)**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $n = 20$ | B1 | Seen or implied |
| $S_{20} = \frac{35000(1-1.04^{20})}{(1-1.04)}$ | M1, A1ft | Needs any $r$, $a=35000$, $n=19$, 20 or 21; A1ft from $n=19$ or $n=21$ but $r$ must be 1.04 |
| $= 1\,042\,000$ | A1 | (4 marks) |

*Note: Failure to round correctly in (b) and (c): penalise once only (first occurrence).*

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9. (a) A geometric series has first term $a$ and common ratio $r$. Prove that the sum of the first $n$ terms of the series is

$$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$

Mr. King will be paid a salary of $\pounds 35000$ in the year 2005 . Mr. King's contract promises a $4 \%$ increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.\\
(b) Find, to the nearest $\pounds 100$, Mr. King's salary in the year 2008.

Mr. King will receive a salary each year from 2005 until he retires at the end of 2024.\\
(c) Find, to the nearest $\pounds 1000$, the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024.\\

\hfill \mbox{\textit{Edexcel C2 2005 Q9 [10]}}
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