Math worksheet on sequences with problems involving linear sequences, nth terms, and arithmetic patterns.
A math worksheet titled "Sequences" with three sections containing problems on finding the nth and 10th terms of linear sequences, finding the nth term of sequences with fractional or negative differences, and finding the first four terms of sequences given their nth term formula.
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Step-by-step solution for: GCSE MATHS sequences worksheet
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Show Answer Key & Explanations
Step-by-step solution for: GCSE MATHS sequences worksheet
Here are the solutions to the sequence problems. I have worked through each one step-by-step to ensure accuracy.
To find the $n^{\text{th}}$ term of a linear sequence, we use the formula:
$$n^{\text{th}} \text{ term} = dn + c$$
Where $d$ is the common difference (the amount it goes up or down by) and $c$ is a constant we find by checking the first term.
a) 6, 10, 14, 18, 22...
* Difference ($d$): $+4$
* Formula: Start with $4n$. For $n=1$, $4(1)=4$. We need 6, so add 2.
* $n^{\text{th}}$ term: $4n + 2$
* $10^{\text{th}}$ term: $4(10) + 2 = 42$
b) 1, 6, 11, 16, 21...
* Difference ($d$): $+5$
* Formula: Start with $5n$. For $n=1$, $5(1)=5$. We need 1, so subtract 4.
* $n^{\text{th}}$ term: $5n - 4$
* $10^{\text{th}}$ term: $5(10) - 4 = 46$
c) 4, 6, 8, 10...
* Difference ($d$): $+2$
* Formula: Start with $2n$. For $n=1$, $2(1)=2$. We need 4, so add 2.
* $n^{\text{th}}$ term: $2n + 2$
* $10^{\text{th}}$ term: $2(10) + 2 = 22$
d) 13, 16, 19, 22, 25...
* Difference ($d$): $+3$
* Formula: Start with $3n$. For $n=1$, $3(1)=3$. We need 13, so add 10.
* $n^{\text{th}}$ term: $3n + 10$
* $10^{\text{th}}$ term: $3(10) + 10 = 40$
e) 7, 8, 9, 10, 11...
* Difference ($d$): $+1$
* Formula: Start with $1n$. For $n=1$, $1(1)=1$. We need 7, so add 6.
* $n^{\text{th}}$ term: $n + 6$
* $10^{\text{th}}$ term: $10 + 6 = 16$
f) 1, 9, 17, 25...
* Difference ($d$): $+8$
* Formula: Start with $8n$. For $n=1$, $8(1)=8$. We need 1, so subtract 7.
* $n^{\text{th}}$ term: $8n - 7$
* $10^{\text{th}}$ term: $8(10) - 7 = 73$
g) 2, 11, 20, 29, 38...
* Difference ($d$): $+9$
* Formula: Start with $9n$. For $n=1$, $9(1)=9$. We need 2, so subtract 7.
* $n^{\text{th}}$ term: $9n - 7$
* $10^{\text{th}}$ term: $9(10) - 7 = 83$
h) 5, 15, 25, 35, 45...
* Difference ($d$): $+10$
* Formula: Start with $10n$. For $n=1$, $10(1)=10$. We need 5, so subtract 5.
* $n^{\text{th}}$ term: $10n - 5$
* $10^{\text{th}}$ term: $10(10) - 5 = 95$
i) 0, 3, 6, 9, 12...
* Difference ($d$): $+3$
* Formula: Start with $3n$. For $n=1$, $3(1)=3$. We need 0, so subtract 3.
* $n^{\text{th}}$ term: $3n - 3$
* $10^{\text{th}}$ term: $3(10) - 3 = 27$
j) -1, 5, 11, 17, 23...
* Difference ($d$): $+6$
* Formula: Start with $6n$. For $n=1$, $6(1)=6$. We need -1, so subtract 7.
* $n^{\text{th}}$ term: $6n - 7$
* $10^{\text{th}}$ term: $6(10) - 7 = 53$
k) -3, 7, 17, 27, 37...
* Difference ($d$): $+10$
* Formula: Start with $10n$. For $n=1$, $10(1)=10$. We need -3, so subtract 13.
* $n^{\text{th}}$ term: $10n - 13$
* $10^{\text{th}}$ term: $10(10) - 13 = 87$
l) -10, -8, -6...
* Difference ($d$): $+2$
* Formula: Start with $2n$. For $n=1$, $2(1)=2$. We need -10, so subtract 12.
* $n^{\text{th}}$ term: $2n - 12$
* $10^{\text{th}}$ term: $2(10) - 12 = 8$
---
a) 12, 10, 8, 6, 4...
* Difference is $-2$. Formula starts with $-2n$.
* Check $n=1$: $-2(1) = -2$. To get 12, add 14.
* Answer: $-2n + 14$ (or $14 - 2n$)
b) 3.5, 4, 4.5, 5, 5.5...
* Difference is $+0.5$. Formula starts with $0.5n$.
* Check $n=1$: $0.5(1) = 0.5$. To get 3.5, add 3.
* Answer: $0.5n + 3$
c) 0.2, 0.4, 0.6...
* Difference is $+0.2$. Formula starts with $0.2n$.
* Check $n=1$: $0.2(1) = 0.2$. This matches the first term exactly.
* Answer: $0.2n$
d) 15, 12, 9, 6, 3...
* Difference is $-3$. Formula starts with $-3n$.
* Check $n=1$: $-3(1) = -3$. To get 15, add 18.
* Answer: $-3n + 18$ (or $18 - 3n$)
e) -4, -8, -12, -16...
* Difference is $-4$. Formula starts with $-4n$.
* Check $n=1$: $-4(1) = -4$. This matches the first term exactly.
* Answer: $-4n$
f) 99, 98, 97...
* Difference is $-1$. Formula starts with $-1n$ (or just $-n$).
* Check $n=1$: $-1$. To get 99, add 100.
* Answer: $100 - n$
---
To do this, simply substitute $n = 1, 2, 3,$ and $4$ into the given formula.
a) $n + 6$
* $n=1: 1+6=7$
* $n=2: 2+6=8$
* $n=3: 3+6=9$
* $n=4: 4+6=10$
* Terms: 7, 8, 9, 10
b) $n + 3$
* $n=1: 1+3=4$
* $n=2: 2+3=5$
* $n=3: 3+3=6$
* $n=4: 4+3=7$
* Terms: 4, 5, 6, 7
c) $n - 5$
* $n=1: 1-5=-4$
* $n=2: 2-5=-3$
* $n=3: 3-5=-2$
* $n=4: 4-5=-1$
* Terms: -4, -3, -2, -1
d) $n + 5$
* $n=1: 1+5=6$
* $n=2: 2+5=7$
* $n=3: 3+5=8$
* $n=4: 4+5=9$
* Terms: 6, 7, 8, 9
e) $4n - 3$
* $n=1: 4(1)-3=1$
* $n=2: 4(2)-3=5$
* $n=3: 4(3)-3=9$
* $n=4: 4(4)-3=13$
* Terms: 1, 5, 9, 13
f) $8n + 11$
* $n=1: 8(1)+11=19$
* $n=2: 8(2)+11=27$
* $n=3: 8(3)+11=35$
* $n=4: 8(4)+11=43$
* Terms: 19, 27, 35, 43
g) $6n - 6$
* $n=1: 6(1)-6=0$
* $n=2: 6(2)-6=6$
* $n=3: 6(3)-6=12$
* $n=4: 6(4)-6=18$
* Terms: 0, 6, 12, 18
h) $5n$
* $n=1: 5(1)=5$
* $n=2: 5(2)=10$
* $n=3: 5(3)=15$
* $n=4: 5(4)=20$
* Terms: 5, 10, 15, 20
i) $-3n + 24$
* $n=1: -3(1)+24=21$
* $n=2: -3(2)+24=18$
* $n=3: -3(3)+24=15$
* $n=4: -3(4)+24=12$
* Terms: 21, 18, 15, 12
Final Answer:
1. nth and 10th terms:
a) $4n + 2$, 42
b) $5n - 4$, 46
c) $2n + 2$, 22
d) $3n + 10$, 40
e) $n + 6$, 16
f) $8n - 7$, 73
g) $9n - 7$, 83
h) $10n - 5$, 95
i) $3n - 3$, 27
j) $6n - 7$, 53
k) $10n - 13$, 87
l) $2n - 12$, 8
2. nth terms:
a) $14 - 2n$
b) $0.5n + 3$
c) $0.2n$
d) $18 - 3n$
e) $-4n$
f) $100 - n$
3. First 4 terms:
a) 7, 8, 9, 10
b) 4, 5, 6, 7
c) -4, -3, -2, -1
d) 6, 7, 8, 9
e) 1, 5, 9, 13
f) 19, 27, 35, 43
g) 0, 6, 12, 18
h) 5, 10, 15, 20
i) 21, 18, 15, 12
1. Find the $n^{\text{th}}$ and $10^{\text{th}}$ term
To find the $n^{\text{th}}$ term of a linear sequence, we use the formula:
$$n^{\text{th}} \text{ term} = dn + c$$
Where $d$ is the common difference (the amount it goes up or down by) and $c$ is a constant we find by checking the first term.
a) 6, 10, 14, 18, 22...
* Difference ($d$): $+4$
* Formula: Start with $4n$. For $n=1$, $4(1)=4$. We need 6, so add 2.
* $n^{\text{th}}$ term: $4n + 2$
* $10^{\text{th}}$ term: $4(10) + 2 = 42$
b) 1, 6, 11, 16, 21...
* Difference ($d$): $+5$
* Formula: Start with $5n$. For $n=1$, $5(1)=5$. We need 1, so subtract 4.
* $n^{\text{th}}$ term: $5n - 4$
* $10^{\text{th}}$ term: $5(10) - 4 = 46$
c) 4, 6, 8, 10...
* Difference ($d$): $+2$
* Formula: Start with $2n$. For $n=1$, $2(1)=2$. We need 4, so add 2.
* $n^{\text{th}}$ term: $2n + 2$
* $10^{\text{th}}$ term: $2(10) + 2 = 22$
d) 13, 16, 19, 22, 25...
* Difference ($d$): $+3$
* Formula: Start with $3n$. For $n=1$, $3(1)=3$. We need 13, so add 10.
* $n^{\text{th}}$ term: $3n + 10$
* $10^{\text{th}}$ term: $3(10) + 10 = 40$
e) 7, 8, 9, 10, 11...
* Difference ($d$): $+1$
* Formula: Start with $1n$. For $n=1$, $1(1)=1$. We need 7, so add 6.
* $n^{\text{th}}$ term: $n + 6$
* $10^{\text{th}}$ term: $10 + 6 = 16$
f) 1, 9, 17, 25...
* Difference ($d$): $+8$
* Formula: Start with $8n$. For $n=1$, $8(1)=8$. We need 1, so subtract 7.
* $n^{\text{th}}$ term: $8n - 7$
* $10^{\text{th}}$ term: $8(10) - 7 = 73$
g) 2, 11, 20, 29, 38...
* Difference ($d$): $+9$
* Formula: Start with $9n$. For $n=1$, $9(1)=9$. We need 2, so subtract 7.
* $n^{\text{th}}$ term: $9n - 7$
* $10^{\text{th}}$ term: $9(10) - 7 = 83$
h) 5, 15, 25, 35, 45...
* Difference ($d$): $+10$
* Formula: Start with $10n$. For $n=1$, $10(1)=10$. We need 5, so subtract 5.
* $n^{\text{th}}$ term: $10n - 5$
* $10^{\text{th}}$ term: $10(10) - 5 = 95$
i) 0, 3, 6, 9, 12...
* Difference ($d$): $+3$
* Formula: Start with $3n$. For $n=1$, $3(1)=3$. We need 0, so subtract 3.
* $n^{\text{th}}$ term: $3n - 3$
* $10^{\text{th}}$ term: $3(10) - 3 = 27$
j) -1, 5, 11, 17, 23...
* Difference ($d$): $+6$
* Formula: Start with $6n$. For $n=1$, $6(1)=6$. We need -1, so subtract 7.
* $n^{\text{th}}$ term: $6n - 7$
* $10^{\text{th}}$ term: $6(10) - 7 = 53$
k) -3, 7, 17, 27, 37...
* Difference ($d$): $+10$
* Formula: Start with $10n$. For $n=1$, $10(1)=10$. We need -3, so subtract 13.
* $n^{\text{th}}$ term: $10n - 13$
* $10^{\text{th}}$ term: $10(10) - 13 = 87$
l) -10, -8, -6...
* Difference ($d$): $+2$
* Formula: Start with $2n$. For $n=1$, $2(1)=2$. We need -10, so subtract 12.
* $n^{\text{th}}$ term: $2n - 12$
* $10^{\text{th}}$ term: $2(10) - 12 = 8$
---
2. Find the $n^{\text{th}}$ term
a) 12, 10, 8, 6, 4...
* Difference is $-2$. Formula starts with $-2n$.
* Check $n=1$: $-2(1) = -2$. To get 12, add 14.
* Answer: $-2n + 14$ (or $14 - 2n$)
b) 3.5, 4, 4.5, 5, 5.5...
* Difference is $+0.5$. Formula starts with $0.5n$.
* Check $n=1$: $0.5(1) = 0.5$. To get 3.5, add 3.
* Answer: $0.5n + 3$
c) 0.2, 0.4, 0.6...
* Difference is $+0.2$. Formula starts with $0.2n$.
* Check $n=1$: $0.2(1) = 0.2$. This matches the first term exactly.
* Answer: $0.2n$
d) 15, 12, 9, 6, 3...
* Difference is $-3$. Formula starts with $-3n$.
* Check $n=1$: $-3(1) = -3$. To get 15, add 18.
* Answer: $-3n + 18$ (or $18 - 3n$)
e) -4, -8, -12, -16...
* Difference is $-4$. Formula starts with $-4n$.
* Check $n=1$: $-4(1) = -4$. This matches the first term exactly.
* Answer: $-4n$
f) 99, 98, 97...
* Difference is $-1$. Formula starts with $-1n$ (or just $-n$).
* Check $n=1$: $-1$. To get 99, add 100.
* Answer: $100 - n$
---
3. Find the first 4 terms
To do this, simply substitute $n = 1, 2, 3,$ and $4$ into the given formula.
a) $n + 6$
* $n=1: 1+6=7$
* $n=2: 2+6=8$
* $n=3: 3+6=9$
* $n=4: 4+6=10$
* Terms: 7, 8, 9, 10
b) $n + 3$
* $n=1: 1+3=4$
* $n=2: 2+3=5$
* $n=3: 3+3=6$
* $n=4: 4+3=7$
* Terms: 4, 5, 6, 7
c) $n - 5$
* $n=1: 1-5=-4$
* $n=2: 2-5=-3$
* $n=3: 3-5=-2$
* $n=4: 4-5=-1$
* Terms: -4, -3, -2, -1
d) $n + 5$
* $n=1: 1+5=6$
* $n=2: 2+5=7$
* $n=3: 3+5=8$
* $n=4: 4+5=9$
* Terms: 6, 7, 8, 9
e) $4n - 3$
* $n=1: 4(1)-3=1$
* $n=2: 4(2)-3=5$
* $n=3: 4(3)-3=9$
* $n=4: 4(4)-3=13$
* Terms: 1, 5, 9, 13
f) $8n + 11$
* $n=1: 8(1)+11=19$
* $n=2: 8(2)+11=27$
* $n=3: 8(3)+11=35$
* $n=4: 8(4)+11=43$
* Terms: 19, 27, 35, 43
g) $6n - 6$
* $n=1: 6(1)-6=0$
* $n=2: 6(2)-6=6$
* $n=3: 6(3)-6=12$
* $n=4: 6(4)-6=18$
* Terms: 0, 6, 12, 18
h) $5n$
* $n=1: 5(1)=5$
* $n=2: 5(2)=10$
* $n=3: 5(3)=15$
* $n=4: 5(4)=20$
* Terms: 5, 10, 15, 20
i) $-3n + 24$
* $n=1: -3(1)+24=21$
* $n=2: -3(2)+24=18$
* $n=3: -3(3)+24=15$
* $n=4: -3(4)+24=12$
* Terms: 21, 18, 15, 12
Final Answer:
1. nth and 10th terms:
a) $4n + 2$, 42
b) $5n - 4$, 46
c) $2n + 2$, 22
d) $3n + 10$, 40
e) $n + 6$, 16
f) $8n - 7$, 73
g) $9n - 7$, 83
h) $10n - 5$, 95
i) $3n - 3$, 27
j) $6n - 7$, 53
k) $10n - 13$, 87
l) $2n - 12$, 8
2. nth terms:
a) $14 - 2n$
b) $0.5n + 3$
c) $0.2n$
d) $18 - 3n$
e) $-4n$
f) $100 - n$
3. First 4 terms:
a) 7, 8, 9, 10
b) 4, 5, 6, 7
c) -4, -3, -2, -1
d) 6, 7, 8, 9
e) 1, 5, 9, 13
f) 19, 27, 35, 43
g) 0, 6, 12, 18
h) 5, 10, 15, 20
i) 21, 18, 15, 12
Parent Tip: Review the logic above to help your child master the concept of sequence math worksheet.
