AQA Level 2 Further Maths · Level 2 Certificate · Chapter 4: Algebra IV · Lesson 7

Sequences and limiting values

About 85 minutesNo calculator73 marks

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Date

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The first two questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Solve $2 x^{2} - 5 x - 12 = 0$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A2
Needed today
Work out the $n$ th term of the linear sequence $17, 11, 5, - 1, \dots$
[2 marks]
Answer
A3
Last lesson
Prove that $(n + 5)^{2} - (n + 1)^{2}$ is a multiple of $8$ for every integer $n$ .
[2 marks]
A4
From chapter 3
$f (x) = 2 x + 1$ and $g (x) = x^{2}$ . Work out $f g (3)$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Work out the $n$ th term of the quadratic sequence $5, 12, 23, 38, 57, \dots$

Hint The second differences are all

4

, so the sequence starts

2 n^{2}

. Take

2 n^{2}

away from each term and find the

n

th term of what is left.

B1
Work out the $n$ th term of the quadratic sequence $1, 9, 23, 43, 69, \dots$
[3 marks]
Answer $\dots\dots\dots\dots$

Example 2

The 1st, 3rd and 5th terms of a quadratic sequence are $6$ , $26$ and $62$ . Work out the $n$ th term of the sequence.

Hint Write the

n

th term as

a n^{2} + bn + c

. Put

n = 1

3

and

5

in to get three equations, then eliminate

c

B2
The 1st, 2nd and 4th terms of a quadratic sequence are $2$ , $9$ and $35$ . Work out the $n$ th term of the sequence.
[4 marks]
Answer $\dots\dots\dots\dots$

Example 3

The $n$ th term of a sequence is $\frac{6 n + 5}{4 n - 1}$ . Prove that the limiting value of the sequence as $n \to \infty$ is $\frac{3}{2}$ .

Hint Divide the top and the bottom by

n

. Then think about what happens to

\frac{5}{n}

and

\frac{1}{n}

n

gets very large.

B3
The $n$ th term of a sequence is $\frac{9 - 2 n}{5 n + 3}$ . Prove that the limiting value of the sequence as $n \to \infty$ is $- 0.4$ .
[3 marks]

Example 4

The $n$ th term of a sequence is $\frac{3 n + 1}{2 n + 3}$ . Which term has the value $1.4$ ? Show that $1.6$ is not a term of the sequence.

Hint Set the

n

th term equal to the value and solve for

n

. A term's position must be a positive whole number.

B4
The $n$ th term of a sequence is $\frac{5 n - 2}{3 n + 1}$ . Work out which term has the value $1.6$ , and show that $2$ is not a term of the sequence.
[4 marks]
Answerthe $\dots\dots$ th term

Example 5

The $n$ th term of a sequence is $n^{2} + 3 n$ . Which two consecutive terms have a sum of $158$ ?

Hint The next term after the

n

th is

(n + 1)^{2} + 3 (n + 1)

. Add the two terms, set the sum equal to

158

and solve.

B5
The $n$ th term of a sequence is $2 n^{2} - n$ . Which two consecutive terms have a sum of $211$ ?
[4 marks]
Answerthe $\dots\dots$ th and $\dots\dots$ th terms

Practice

Level 2

C1
A linear sequence starts $4, 7.5, 11, 14.5, \dots$ Work out the value of the first term that is greater than $300$ .
[3 marks]
Answer
C2
A linear sequence is $a, a + 3 b, a + 6 b, a + 9 b, \dots$ The 2nd term is $23$ and the 5th term is $59$ . Work out the values of $a$ and $b$ , and the $n$ th term of the sequence.
[3 marks]
Answer $a = \dots\dots b = \dots\dots n th term = \dots\dots$

C3
Work out the $n$ th term of the linear sequence $3, 7, 11, 15, \dots$ Hence work out the $n$ th term of the sequence $9, 49, 121, 225, \dots$ in the form $a n^{2} + bn + c$ .
[3 marks]
Answer $\dots\dots\dots\dots$
C4
The $n$ th term of a sequence is $n^{2} - 16 n + 67$ . Prove that every term of the sequence is positive.
[3 marks]
C5
A sequence starts with $3$ . Each term after the first is found by multiplying the previous term by $3$ and then subtracting $4$ . Work out the 4th term. Explain why no term of the sequence is even.
[3 marks]
Answer4th term $= \dots\dots$
C6
Work out the limiting value, as $n \to \infty$ , of the sequence with $n$ th term $\frac{6 n ^{2} - 5 n}{3 n ^{2} + 2 n + 7}$ .
[2 marks]
Answer

Exam-style questions

AQA exam style

D1
Mia says, "The sequence with $n$ th term $\frac{5 n + 2}{n + 1}$ never gets close to $5$ , so $4.9$ is not one of its terms."
$n$ is a positive integer.
(a)
A term of the sequence has a value of $4.7$ . Work out the value of $n$ .
[2 marks]
Answer $n = \dots\dots$
(b)
Write down the limiting value of the sequence as $n \to \infty$ .
[1 mark]
Answer
(c)
Show that Mia is wrong.
[2 marks]
Total for D1: 5 marks

D2
For sequence A, $n$ th term $= \frac{11 n + 40}{n}$ . For sequence B, $n$ th term $= 2 n$ .
(a)
The $k$ th term of sequence A equals the $k$ th term of sequence B. Work out the value of $k$ . You must show your working.
[4 marks]
Answer $k = \dots\dots$
(b)
Write down the limiting value of sequence A as $n \to \infty$ .
[1 mark]
Answer
Total for D2: 5 marks
D3
The first five terms of a quadratic sequence are $4, 3, 0, - 5, - 12$
(a)
Work out an expression for the $n$ th term.
[3 marks]
Answer $\dots\dots\dots\dots$
(b)
Work out which term of the sequence is equal to $- 192$ .
[2 marks]
Answerthe $\dots\dots$ th term
Total for D3: 5 marks
D4
The $n$ th term of a sequence is $\frac{an + 6}{bn - 1}$ , where $a$ and $b$ are positive integers. The 1st term of the sequence is $6$ . The limiting value of the sequence as $n \to \infty$ is $3$ . Work out the value of the 2nd term of the sequence.
[4 marks]
Answer $\dots\dots$
D5
The $n$ th term of a sequence is $T_{n} = \frac{2 n + 1}{n + 2}$ .
(a)
Show that $T_{n + 1} - T_{n} = \frac{3}{( n + 2 ) ( n + 3 )}$ .
[3 marks]
(b)
Hence explain why each term of the sequence is greater than the term before it.
[1 mark]
(c)
Work out the first value of $n$ for which $T_{n + 1} - T_{n} < 0.01$ .
[2 marks]
Answer $n = \dots\dots$
(d)
Write down the limiting value of $T_{n}$ as $n \to \infty$ .
[1 mark]
Answer
Total for D5: 7 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The $n$ th term of a sequence is $\frac{3 n + 7}{n + 1}$ . Find every term of the sequence that is a whole number, and prove that there are no others.
[4 marks]
Hint 1 · What to try
Write the $n$ th term as $3$ plus a fraction.
Hint 2 · The first line
$3 n + 7 = 3 (n + 1) + 4$ , so the $n$ th term is $3 + \frac{4}{n + 1}$ .
Hint 3 · The full method
It is a whole number only when $n + 1$ divides $4$ . With $n \geq 1$ , $n + 1 = 2$ or $4$ , so $n = 1$ (term $5$ ) or $n = 3$ (term $4$ ). For $n \geq 4$ , $0 < \frac{4}{n + 1} < 1$ , so no other term is whole.
Answer $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the $n$ th term of a linear or a quadratic sequence.
I can use the $n$ th term to find a term's position, or show a value is not a term.
I can find and prove the limiting value of a sequence as $n \to \infty$ .
I can solve problems that combine two terms or two sequences.

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