Pearson Edexcel IAL Mathematics P2.4 Sequences & series Question Bank

Practise sequences, series and binomial expansion through recurrence terms, AP and GP sums, periodic behaviour and coefficient finding.

Exam points

generate u_n terms from a recurrence and use a given term or sum to solve for constants
apply AP or GP formulae to find n, S_n or the sum to infinity when |r|<1
use nCr terms in a binomial expansion and link a chosen x value to an approximation

Covered syllabus points

P2.4.1 - Sequences,P2.4.2 - Arithmetic sequences and series P2.4.3 - Increasing sequences, decreasing sequences and periodic P2.4.4 - Geometric sequences and series P2.4.5 - Binomial expansion for positive integer powers

[Maximum number: 5]

A sequence is defined by

\begin{aligned} u_{1} & =6 \\ u_{n+1} & =k u_{n}+3 \end{aligned}

where k is a positive constant.

(a)

Find, in terms of k, an expression for $u_{3}$

Given that $\sum_{n=1}^{3} u_{n}=117$

[ 2 ]

Substitute $u_2=6k+3$ into $u_3=ku_2+3$ :
$u_3=k(6k+3)+3=6k^2+3k+3.$
M1 A1

(b)

find the value of k.

[ 3 ]

M1: Sets their $u_{1}+u_{2}+u_{3}=117$ to produce an equation in just k.
dM1: Solves a 3TQ by any valid method to find at least one value for k.
A1: $\quad k=\frac{7}{2}$ ONLY

Figure for Answer to Question (b) — Edexcel A-Level Mathematics AS

[Maximum number: 7]

In a large theatre there are n rows of seats, where n is a constant.

The number of seats in the first row is a, where a is a constant.
In each subsequent row there are 4 more seats than in the previous row so that
- in the 2 nd row there are (a+4) seats
- in the 3 rd row there are (a+8) seats
- the number of seats in each row form an arithmetic sequence

Given that the total number of seats in the first 10 rows is 360

(a)

find the value of a.

Given also that the total number of seats in the n rows is 2146

[ 2 ]

M1: Uses $S_{10}=360$ with a correct formula to set up a linear equation in " a ". E.g. $\frac{10}{2}\{2 \times a+9 \times 4\}=360$ Alternatively uses the sum of all 10 terms

a+(a+4)+(a+8)+(a+12)+(a+16)+(a+20)+(a+24)+(a+28)+(a+32)+(a+36)=360

A1: a=18 following a correct equation

(b)

show that

n^{2}+8 n-1073=0

[ 2 ]

$S_n=2146$ $\frac{n}{2}\{2(18)+(n-1)4\}=2146$ $n^2+8n-1073=0$

M1
A1*

(c)

Hence

[ 3 ]

(i)

state the number of rows of seats in the theatre,

[ 1 ]

(ii)

find the maximum number of seats in any one row.

[ 2 ]

$18+(29-1)\times4=130$
Maximum number of seats =130.

M1
A1

[Maximum number: 8]

(a)

Find the value of

\sum_{r=1}^{\infty} 6 \times(0.25)^{r}

(3)

[ 3 ]

r=0.25

B1

$S_{\infty}=\frac{1.5}{1-0.25}=2$

M1 A1

(3)

(b)

A sequence $u_{1}, u_{2}, u_{3}, \ldots$ is defined by

\begin{aligned} u_{1} & =3 \\ u_{n+1} & =\frac{u_{n}-3}{u_{n}-2} \quad n \in \mathbb{N} \end{aligned}

[ 5 ]

(i)

Show that this sequence is periodic.

[ 2 ]

$\quad\left(u_{1}=3,\right) u_{2}=0, u_{3}=\frac{3}{2}, u_{4}=3, \ldots$

M1

$\left(u_{1}=3\right), u_{2}=0, u_{3}=\frac{3}{2}, u_{4}=3, \ldots$ and e.g. states that $u_{4}=u_{1}$ so sequence will repeat (is
periodic)

A1

(2)

(ii)

State the order of this sequence.

[ 1 ]

(iii)

Hence find

\sum_{n=1}^{70} u_{n}

[ 2 ]

The sequence repeats with period 3, with one cycle
$3,\ 0,\ \frac32.$
Since $70=23\cdot3+1$ ,
$\sum_{n=1}^{70}u_n=23\left(3+0+\frac32\right)+3=106.5=\frac{213}{2}.$
M1 A1

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Covered syllabus points

P2.4 - Sequences and series question 1

Question (a)

Question (b)

P2.4 - Sequences and series question 2

Question (a)

Question (b)

Question (c)

Question (i)

Question (ii)

P2.4 - Sequences and series question 3

Question (a)

Question (b)

Question (i)

Question (ii)

Question (iii)