C1 June 2006 Q4

EdexcelOld spec5 marksSequences & Series

4. A sequence \(a_1, a_2, a_3, \ldots\) is defined by\[\begin{aligned} a_1 &= 3, \\ a_{n+1} &= 3a_n - 5, \quad n \geqslant 1. \end{aligned}\]

(a) Find the value of \(a_2\) and the value of \(a_3\). (2)

(b) Calculate the value of \(\displaystyle\sum_{r=1}^{5} a_r\). (3)

Mark scheme (a)

Scheme	Marks
\(a_2 = 4\)	B1
\(a_3 = 3\times a_2 - 5 = 7\)	B1f.t.
	(2)

Notes

2^nd B1f.t. Follow through their \(a_2\) but it must be a value. \(3\times 4 - 5\) is B0
Give wherever it is first seen.

Mark scheme (b)

Scheme	Marks
\(a_4 = 3a_3 - 5\,(= 16)\) and \(a_5 = 3a_4 - 5\,(= 43)\)	M1
\(3 + 4 + 7 + 16 + 43\)	M1
\(= 73\)	A1c.a.o.
	(3)
	(5 marks)

Notes

1^st M1 For two further attempts to use of \(a_{n+1} = 3a_n - 5\), wherever seen.
Condone arithmetic slips

2^nd M1 For attempting to add 5 relevant terms (i.e. terms derived from an attempt to use the recurrence formula) or an expression.
Follow through their values for \(a_2 - a_5\)
Use of formulae for arithmetic series is M0A0 but could get 1^st M1 if \(a_4\) and \(a_5\) are correctly attempted.