A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 3: Sequences and series · Lesson 4

Sigma notation and recurrence relations

115 marks

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Date

Do Now

Quick questions to start. They use what you need today.

A1
Needed today
Evaluate $r = 1 \sum 4 (2 r + 1)$ .
[2 marks]
Answer
A2
Common slip
How many terms are there in $r = 5 \sum 30 r^{2}$ ?
[2 marks]
Answer
A3
Needed today
A sequence has $u_{1} = 4$ and $u_{n + 1} = 3 u_{n} - 2$ . Find $u_{3}$ .
[2 marks]
Answer
A4
From chapter 3
Find the sum of the first $6$ terms of the geometric series $5 + 15 + 45 + \dots$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Evaluate $r = 1 \sum 30 (4 r - 1)$ .

Hint Find the first term, the last term and the number of terms. It is an arithmetic series.

B1
Evaluate $r = 10 \sum 50 (100 - 2 r)$ .
[4 marks]
Answer

Example 2

Evaluate $r = 1 \sum 8 3 \times 2^{r}$ .

Hint Write out the first term. When

r = 1

the term is

6

, not

3

B2
Evaluate $r = 0 \sum 9 5 (0.5)^{r}$ , giving your answer as an exact fraction.
[4 marks]
Answer

Example 3

Evaluate $r = 1 \sum 20 (2^{r} + 3 r)$ .

Hint Split it into

\sum 2^{r}

(geometric) and

\sum 3 r

(arithmetic). Find each, then add.

B3
Evaluate $r = 1 \sum 10 (3^{r} - 4 r + 1)$ .
[5 marks]
Answer

Example 4

A sequence has $u_{1} = 2$ and $u_{n + 1} = k u_{n} + 3$ . Given $u_{3} = 23$ , find the possible values of $k$ .

Hint Write

u_{2}

and then

u_{3}

in terms of

k

. Set

u_{3} = 23

and solve the quadratic.

B4
A sequence has $u_{1} = 3$ and $u_{n + 1} = k u_{n} - 4$ . Given $u_{3} = 11$ , find the possible values of $k$ .
[5 marks]
Answer

Example 5

A sequence has $u_{1} = 1$ and $u_{n + 1} = 4 - u_{n}$ . Find $r = 1 \sum 25 u_{r}$ .

Hint Write out the first few terms. When a term repeats, the sequence repeats. Count the full cycles in

25

terms.

B5
A sequence has $u_{1} = 2$ and $u_{n + 1} = \frac{1}{1 - u _{n}}$ . Find $r = 1 \sum 50 u_{r}$ .
[5 marks]
Answer

Practice

A-Level

C1
A sequence has $u_{1} = 1$ and $u_{n + 1} = u_{n} + \frac{1}{u _{n}}$ . Find $u_{2}$ , $u_{3}$ and $u_{4}$ , and prove that the sequence is increasing.
[4 marks]
Answer $u_{2} = \dots$ , $u_{3} = \dots$ , $u_{4} = \dots$

C2
Show that $r = 1 \sum n (6 r - 1) = 3 n^{2} + 2 n$ , and find the least $n$ for which the sum is greater than $2000$ .
[5 marks]
Answer
C3
A sequence has $u_{1} = 2$ and $u_{n + 1} = 0.5 u_{n} + 6$ . Find $u_{2}$ , $u_{3}$ and $u_{4}$ . The sequence tends to a limit $L$ . Find $L$ .
[5 marks]
Answer
C4
A sequence has $u_{1} = 5$ and $u_{n + 1} = 3 u_{n} - 8$ . A student writes: " $L = 3 L - 8$ gives $L = 4$ , so the sequence tends to $4$ ." Find $u_{2}$ , $u_{3}$ and $u_{4}$ , and explain why the student is wrong.
[4 marks]
Answer $u_{2} = \dots$ , $u_{3} = \dots$ , $u_{4} = \dots$
C5
Find $r = 1 \sum \infty 8 (- \frac{1}{2})^{r}$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A sequence is given by $u_{1} = 2$ , $u_{n + 1} = a u_{n} + b$ , where $a$ and $b$ are constants. Given $u_{2} = 7$ and $u_{3} = 22$ :
(a)
find $a$ and $b$ .
[3 marks]
Answer
(b)
Find $u_{5}$ .
[2 marks]
Answer
(c)
Find $r = 1 \sum 5 u_{r}$ .
[2 marks]
Answer
(d)
Show that $u_{n} = \frac{5 \times 3 ^{n - 1} - 1}{2}$ gives $u_{1} = 2$ and satisfies $u_{n + 1} = 3 u_{n} + 1$ .
[3 marks]
Answer
Total for D1: 10 marks

D2
Zoe says: " $r = 4 \sum 12 (3 r - 1)$ has $8$ terms, starting at $11$ and going up by $3$ , so it equals $\frac{8}{2} (22 + 21) = 172$ ."
Zoe is wrong.
(a)
Find the correct value of the sum, and explain the mistake Zoe made.
[3 marks]
Answer
(b)
Given that $r = 4 \sum n (3 r - 1) = 1350$ , find $n$ .
[5 marks]
Answer
Total for D2: 8 marks
D3
Given that $r = 1 \sum n (2 r + k) = n^{2} + 7 n$ for all positive integers $n$ , find the constant $k$ , and hence find $r = 11 \sum 20 (2 r + k)$ .
[7 marks]
Answer
D4
A sequence has $u_{1} = 4$ and $u_{n + 1} = \frac{2 u _{n} + k}{u _{n} - 2}$ , where $k$ is a constant and $k \neq = - 4$ .
(a)
Find $u_{2}$ in terms of $k$ .
[2 marks]
Answer
(b)
Show that $u_{3} = 4$ .
[3 marks]
Answer
(c)
Given $r = 1 \sum 30 u_{r} = 225$ , find $k$ .
[3 marks]
Answer
(d)
Find the value of $k$ for which every term of the sequence is $4$ .
[2 marks]
Answer
Total for D4: 10 marks
D5
The sum $S_{n} = r = 1 \sum n 5 \times 2^{r}$ .
(a)
Show that $S_{n} = 10 (2^{n} - 1)$ .
[3 marks]
Answer
(b)
Find the least $n$ for which $S_{n} > 1 0^{6}$ .
[3 marks]
Answer
(c)
Given $r = n + 1 \sum 2 n 5 \times 2^{r} = 652800$ , find $n$ .
[3 marks]
Answer
Total for D5: 9 marks
D6
A sequence has $u_{1} = 3$ and $u_{n + 1} = k u_{n} + 5$ , where $k$ is a constant.
(a)
Show that $r = 1 \sum 3 u_{r} = 3 k^{2} + 8 k + 13$ .
[3 marks]
(b)
Given that $r = 1 \sum 3 u_{r} = 16$ , find the two possible values of $k$ .
[3 marks]
Answer
(c)
Hence find the value of $k$ for which the sequence converges, and find its limit. Explain why the sequence does not converge for the other value.
[3 marks]
Answer
Total for D6: 9 marks

Extension

Stretch

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

E1
A sequence has $u_{n + 1} = u_{n}^{2} - 6$ . Given that $u_{3} = u_{1}$ but $u_{2} \neq = u_{1}$ , find the possible values of $u_{1}$ in exact form, and find $r = 1 \sum 100 u_{r}$ .
[9 marks]
Hint 1 · What to try
Write $u_{1} = x$ . Then $u_{3} = (x^{2} - 6)^{2} - 6 = x$ , a quartic.
Hint 2 · The first line
The values with $u_{2} = u_{1}$ solve $x^{2} - 6 = x$ , so $x^{2} - x - 6$ is a factor of the quartic.
Hint 3 · The full method
The other factor is $x^{2} + x - 5$ . Use it to write $u_{1} + u_{2}$ simply.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can count the terms in $\sum_{r = a}^{b}$ and evaluate arithmetic and geometric sums written in sigma notation.
I can split a sum into parts and find a sum that does not start at $r = 1$ .
I can use a recurrence relation to find terms and unknown constants.
I can spot a periodic sequence and use it to find a long sum.

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