Pack A — Fluency

Mathematics

Year 10 · Sequences

Pack A · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

1.
Write down the next two terms in the sequence: $3,\ 7,\ 11,\ 15,\ \ldots$ .
2.
Write down the next two terms in the sequence: $2,\ 6,\ 18,\ 54,\ \ldots$ .
3.
State whether the sequence $4,\ 8,\ 12,\ 16,\ \ldots$ is arithmetic, geometric, or neither.
4.
For the sequence $15,\ 11,\ 7,\ 3,\ \ldots$ , state the common difference $d$ .
5.
For the sequence $64,\ 32,\ 16,\ 8,\ \ldots$ , state the common ratio $r$ .
6.
A sequence begins $7,\ 10,\ 13,\ 16,\ \ldots$ . Write down (a) $u_1$ and (b) $u_3$ .
7.
A sequence has first term $u_1 = 5$ and term-to-term rule "add $3$ ". Write down the first four terms.
8.
A sequence has $u_1 = 2$ and term-to-term rule "multiply by $3$ ". Write down the first four terms.
9.
The $n$ th term of a sequence is $u_n = 3n + 2$ . Find $u_1$ and $u_5$ .
10.
Find the 6th term of the sequence $2,\ 5,\ 8,\ 11,\ \ldots$ .

SilverQuestions 11–20

11.
Find the $n$ th term formula for the arithmetic sequence $5,\ 8,\ 11,\ 14,\ \ldots$ .
12.
Find the $n$ th term formula for the sequence $20,\ 17,\ 14,\ 11,\ \ldots$ .
13.
For the arithmetic sequence with $u_1 = 7$ and $d = 4$ , find $u_{{n}}$ .
14.
For the sequence with $n$ th term $u_n = 4n + 1$ , find which term equals $81$ .
15.
A geometric sequence has $u_1 = 3$ and $r = 2$ . Find $u_5$ .
16.
A sequence is defined recursively by $u_1 = 4$ and $u_{{n+1}} = u_n + 5$ . Find the first four terms.
17.
A sequence is defined by $u_1 = 5$ and $u_{{n+1}} = 2 u_n$ . Find $u_4$ .
18.
A pattern uses $4$ matchsticks for shape 1, and each new shape adds $3$ more. How many matchsticks for shape 10?
19.
An arithmetic sequence has $u_3 = 11$ and $u_8 = 31$ . Find the common difference.
20.
Find the $n$ th term formula for the geometric sequence $4,\ 12,\ 36,\ 108,\ \ldots$ .

GoldQuestions 21–30

21.
An arithmetic sequence has $u_5 = 17$ and $u_{{n}} = 47$ where $n = 15$ . Find $u_1$ and $d$ .
22.
The $n$ th term of a sequence is $u_n = 7n + 3$ . Find the first term greater than $100$ .
23.
A sequence is defined by $u_1 = 5$ and $u_{{n+1}} = 2u_n - 3$ . Find $u_4$ .
24.
A geometric sequence has $u_2 = 6$ and $u_4 = 54$ (both positive). Find $u_1$ and $r$ .
25.
A sequence is given by $1,\ 4,\ 9,\ 16,\ 25,\ \ldots$ . Find the next term and describe the pattern.
26.
A sequence is defined by $u_1 = 1$ , $u_2 = 2$ and $u_{{n+1}} = u_n + u_{{n-1}}$ . Find $u_6$ .
27.
Which term of the sequence $u_n = 8n - 5$ equals $99$ ?
28.
The first three terms of a sequence are $7,\ 4,\ 1,\ -2,\ \ldots$ . (a) State the type. (b) Find $u_n$ .
29.
Find the sum of the first $10$ terms of the arithmetic sequence with $u_1 = 3$ and $d = 5$ .
30.
A geometric sequence has $u_1 = 16$ and $r = \dfrac{1}{2}$ . Find $u_4$ .

PlatinumQuestions 31–40

31.
An arithmetic sequence has $u_4 = 14$ and $u_{{n}} = 44$ where $n = 14$ . Find $u_n$ as a formula.
32.
Find the sum of all multiples of $7$ between 1 and $100$ inclusive.
33.
A geometric sequence has $u_3 = 12$ and $u_6 = 96$ (both positive). Find $u_1$ and $r$ .
34.
Sequence A has $u_n = 5n + 2$ and sequence B has $u_n = 8n - 7$ . For which value of $n$ do they have the same term?
35.
A sequence is defined by $u_1 = 8$ and $u_{{n+1}} = \dfrac{u_n}{2} + 3$ . Find $u_4$ .
36.
Find the sum of the first $6$ terms of the geometric sequence with $u_1 = 2$ and $r = 3$ .
37.
The first four terms of a sequence are $2,\ 6,\ 12,\ 20,\ \ldots$ . Find the next term and a quadratic formula for $u_n$ .
38.
A bacterial colony of $50$ cells doubles every hour. (a) Write the number of cells $u_n$ after $n$ hours. (b) After how many full hours will the population exceed $1000$ ?
39.
Maya saves £ $5$ in week 1 and £ $3$ more each week than the previous week. In which week does her **weekly** saving first exceed £ $50$ ?
40.
A sequence is defined by $u_1 = 1$ and $u_{{n+1}} = u_n + 2n + 1$ . Find $u_5$ .