C1 January 2012 Q9

EdexcelOld spec9 marksSequences & Series

9. A company offers two salary schemes for a 10-year period, Year 1 to Year 10 inclusive.

Scheme 1:	Salary in Year 1 is £\(P\). Salary increases by £\((2T)\) each year, forming an arithmetic sequence.
Scheme 2:	Salary in Year 1 is £\((P + 1800)\). Salary increases by £\(T\) each year, forming an arithmetic sequence.

(a) Show that the total earned under Salary Scheme 1 for the 10-year period is\[\text{£}(10P + 90T)\] (2)

For the 10-year period, the total earned is the same for both salary schemes.

(b) Find the value of \(T\). (4)

For this value of \(T\), the salary in Year 10 under Salary Scheme 2 is £29 850

Mark scheme (a)

Scheme	Marks
\(S_{10} = \dfrac{10}{2}\left[2P + 9\times 2T\right]\) or \(\dfrac{10}{2}\left(P + [P + 18T]\right)\) e.g. \(5[2P + 18T]\)	M1
\(= (\text{£})\,(10P + 90T)\) or \((\text{£})\ 10P + 90T \qquad (*)\)	A1cso
	(2)

Notes

M1: for identifying \(a = P\) or \(d = 2T\) and attempt at \(S_{10}\). Using \(n = 10\) and one of \(a\) or \(d\) correct. Must see evidence for M mark, at least one line before the answer.

A1cso: for simplifying to given answer. No incorrect working seen.
Do not penalise missing end bracket in working eg \(5(2P + 18T\)

List: M1A1 for a full list seen (with + signs or written in columns) and no incorrect working seen. Any missing terms is M0A0

Mark scheme (b)

Scheme	Marks
Scheme 2: \(S_{10} = \dfrac{10}{2}\left[2(P + 1800) + 9T\right] = \left\{10P + 18000 + 45T\right\}\)	M1A1
\(10P + 90T = 10P + 18000 + 45T\)	M1
\(90T = 18000 + 45T\) \(T = 400\) (only)	A1
	(4)

Notes

1^st M1: for attempting \(S_{10}\) for scheme 2 (allow missing (…) brackets e.g. \(2P + 1800 + 9T\))
Using \(n = 10\) and at least one of \(a\) or \(d\) correct.

1^st A1: for a correct expression for \(S_{10}\) using scheme 2 (needn’t be multiplied out)

List: Allow M1A1 if they reach \(10P + 18000 + 45T\) with no incorrect working seen
\(10P + 18000 + 45T\) with no working is M1A1

2^nd M1: for forming an equation using the two sums that would enable \(P\) to be eliminated.
Follow through their expressions provided \(P\) would disappear.

2^nd A1: for \(T = 400\) Answer only (4/4)

Mark scheme (c)

Scheme	Marks
Scheme 2, Year 10 salary: \([a + (n - 1)d =](P + 1800) + 9T\)	B1ft
\(P + 1800 + \text{“}3600\text{”} = 29850\)	M1
\(P = (\text{£})\ \underline{24450}\)	A1
	(3)
	(9 marks)

Notes

B1: for using \(u_{10}\) for scheme 2. Can be \(9T\) or follow through their value of \(T\)

M1: for forming an equation based on \(u_{10}\) for scheme 2 and using 29850 and their value of \(T\)

A1: for 24450 seen Answer only (3/3)

MR: If they misread scheme 2 as scheme 1 in part (c) apply MR rule and award B0M1A0 max for an equation based on \(u_{10}\) for scheme 1 and using 29850 and their value of \(T\)