GCSE MATHS sequences worksheet - Free Printable

Let’s solve each part step by step.

---

Question 1: Find the nᵗʰ and 10ᵗʰ term of the following linear sequences

A linear sequence has a constant difference between terms. The general formula is:

> nᵗʰ term = a + (n - 1)d

Where:
- `a` = first term
- `d` = common difference
- `n` = position in the sequence

We’ll find `a`, `d`, then plug into the formula to get the nᵗʰ term, then substitute n=10 for the 10ᵗʰ term.

---

a) 6, 10, 14, 18, 22...

- a = 6
- d = 10 - 6 = 4
- nᵗʰ term = 6 + (n - 1)*4 = 6 + 4n - 4 = 4n + 2
- 10ᵗʰ term = 4*10 + 2 = 42

✔ Check: 6,10,14,18,22,26,30,34,38,42 → yes, 10th is 42.

---

b) 1, 6, 11, 16, 21...

- a = 1
- d = 5
- nᵗʰ term = 1 + (n - 1)*5 = 1 + 5n - 5 = 5n - 4
- 10ᵗʰ term = 5*10 - 4 = 46

✔ Check: 1,6,11,16,21,26,31,36,41,46 → correct.

---

c) 4, 6, 8, 10...

- a = 4
- d = 2
- nᵗʰ term = 4 + (n - 1)*2 = 4 + 2n - 2 = 2n + 2
- 10ᵗʰ term = 2*10 + 2 = 22

✔ Check: 4,6,8,10,12,14,16,18,20,22 → correct.

---

d) 13, 16, 19, 22, 25...

- a = 13
- d = 3
- nᵗʰ term = 13 + (n - 1)*3 = 13 + 3n - 3 = 3n + 10
- 10ᵗʰ term = 3*10 + 10 = 40

✔ Check: 13,16,19,22,25,28,31,34,37,40 → correct.

---

e) 7, 8, 9, 10, 11...

- a = 7
- d = 1
- nᵗʰ term = 7 + (n - 1)*1 = 7 + n - 1 = n + 6
- 10ᵗʰ term = 10 + 6 = 16

✔ Check: 7,8,9,10,11,12,13,14,15,16 → correct.

---

f) 1, 9, 17, 25...

- a = 1
- d = 8
- nᵗʰ term = 1 + (n - 1)*8 = 1 + 8n - 8 = 8n - 7
- 10ᵗʰ term = 8*10 - 7 = 73

✔ Check: 1,9,17,25,33,41,49,57,65,73 → correct.

---

g) 2, 11, 20, 29, 38...

- a = 2
- d = 9
- nᵗʰ term = 2 + (n - 1)*9 = 2 + 9n - 9 = 9n - 7
- 10ᵗʰ term = 9*10 - 7 = 83

✔ Check: 2,11,20,29,38,47,56,65,74,83 → correct.

---

h) 5, 15, 25, 35, 45...

- a = 5
- d = 10
- nᵗʰ term = 5 + (n - 1)*10 = 5 + 10n - 10 = 10n - 5
- 10ᵗʰ term = 10*10 - 5 = 95

✔ Check: 5,15,25,35,45,55,65,75,85,95 → correct.

---

i) 0, 3, 6, 9, 12...

- a = 0
- d = 3
- nᵗʰ term = 0 + (n - 1)*3 = 3n - 3
- 10ᵗʰ term = 3*10 - 3 = 27

✔ Check: 0,3,6,9,12,15,18,21,24,27 → correct.

---

j) -1, 5, 11, 17, 23...

- a = -1
- d = 6
- nᵗʰ term = -1 + (n - 1)*6 = -1 + 6n - 6 = 6n - 7
- 10ᵗʰ term = 6*10 - 7 = 53

✔ Check: -1,5,11,17,23,29,35,41,47,53 → correct.

---

k) -3, 7, 17, 27, 37...

- a = -3
- d = 10
- nᵗʰ term = -3 + (n - 1)*10 = -3 + 10n - 10 = 10n - 13
- 10ᵗʰ term = 10*10 - 13 = 87

✔ Check: -3,7,17,27,37,47,57,67,77,87 → correct.

---

l) -10, -8, -6, -4...

- a = -10
- d = 2
- nᵗʰ term = -10 + (n - 1)*2 = -10 + 2n - 2 = 2n - 12
- 10ᵗʰ term = 2*10 - 12 = 8

✔ Check: -10,-8,-6,-4,-2,0,2,4,6,8 → correct.

---

Question 2: Find the nᵗʰ term of the following linear sequences

Same method — find `a` and `d`, then use formula: a + (n - 1)d

---

a) 12, 10, 8, 6, 4...

- a = 12
- d = -2
- nᵗʰ term = 12 + (n - 1)(-2) = 12 - 2n + 2 = -2n + 14

✔ Check: n=1 → 12, n=2→10, n=3→8 → correct.

---

b) 3.5, 4, 4.5, 5, 5.5...

- a = 3.5
- d = 0.5
- nᵗʰ term = 3.5 + (n - 1)(0.5) = 3.5 + 0.5n - 0.5 = 0.5n + 3

Or write as fraction: (1/2)n + 3

✔ Check: n=1 → 3.5, n=2→4, n=3→4.5 → correct.

---

c) 0.2, 0.4, 0.6, 0.8...

- a = 0.2
- d = 0.2
- nᵗʰ term = 0.2 + (n - 1)(0.2) = 0.2 + 0.2n - 0.2 = 0.2n

Or (1/5)n

✔ Check: n=1→0.2, n=2→0.4 → correct.

---

d) 15, 12, 9, 6, 3...

- a = 15
- d = -3
- nᵗʰ term = 15 + (n - 1)(-3) = 15 - 3n + 3 = -3n + 18

✔ Check: n=1→15, n=2→12, n=3→9 → correct.

---

e) ?, -4, -8, -12, -16...

First term missing. But we can see from second term onward:

From -4 to -8: d = -4

So if term 2 = -4, then term 1 = -4 - (-4)? Wait — let's think.

If d = -4, and term 2 = -4, then term 1 = term 2 - d? No:

Actually: term₂ = term₁ + d → so term₁ = term₂ - d

But d = -4 → term₁ = -4 - (-4) = 0? Let’s check:

Sequence: 0, -4, -8, -12, -16 → yes! That works.

So a = 0, d = -4

nᵗʰ term = 0 + (n - 1)(-4) = -4(n - 1) or -4n + 4

✔ Check: n=1→0, n=2→-4, n=3→-8 → correct.

---

f) 99, 98, 97, 96...

- a = 99
- d = -1
- nᵗʰ term = 99 + (n - 1)(-1) = 99 - n + 1 = -n + 100

✔ Check: n=1→99, n=2→98 → correct.

---

Question 3: Find the first 4 terms of each linear sequence whose nᵗʰ term is given

Just plug in n = 1, 2, 3, 4 into the formula.

---

a) n + 6

- n=1: 1+6=7
- n=2: 2+6=8
- n=3: 3+6=9
- n=4: 4+6=10
→ 7, 8, 9, 10

---

b) n + 3

- n=1: 4
- n=2: 5
- n=3: 6
- n=4: 7
→ 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
→ -4, -3, -2, -1

---

d) n + 5

- n=1: 6
- n=2: 7
- n=3: 8
- n=4: 9
→ 6, 7, 8, 9

---

e) 4n - 3

- n=1: 4-3=1
- n=2: 8-3=5
- n=3: 12-3=9
- n=4: 16-3=13
→ 1, 5, 9, 13

---

f) 8n + 11

- n=1: 8+11=19
- n=2: 16+11=27
- n=3: 24+11=35
- n=4: 32+11=43
→ 19, 27, 35, 43

---

g) 6n - 6

- n=1: 6-6=0
- n=2: 12-6=6
- n=3: 18-6=12
- n=4: 24-6=18
→ 0, 6, 12, 18

---

h) 5n

- n=1: 5
- n=2: 10
- n=3: 15
- n=4: 20
→ 5, 10, 15, 20

---

i) -3n + 24

- n=1: -3+24=21
- n=2: -6+24=18
- n=3: -9+24=15
- n=4: -12+24=12
→ 21, 18, 15, 12

---

All answers verified with step-by-step calculation.

Final Answer:

Question 1:
a) nᵗʰ: 4n + 2; 10ᵗʰ: 42
b) nᵗʰ: 5n - 4; 10ᵗʰ: 46
c) nᵗʰ: 2n + 2; 10ᵗʰ: 22
d) nᵗʰ: 3n + 10; 10ᵗʰ: 40
e) nᵗʰ: n + 6; 10ᵗʰ: 16
f) nᵗʰ: 8n - 7; 10ᵗʰ: 73
g) nᵗʰ: 9n - 7; 10ᵗʰ: 83
h) nᵗʰ: 10n - 5; 10ᵗʰ: 95
i) nᵗʰ: 3n - 3; 10ᵗʰ: 27
j) nᵗʰ: 6n - 7; 10ᵗʰ: 53
k) nᵗʰ: 10n - 13; 10ᵗʰ: 87
l) nᵗʰ: 2n - 12; 10ᵗʰ: 8

Question 2:
a) -2n + 14
b) 0.5n + 3 (or ½n + 3)
c) 0.2n (or ⅕n)
d) -3n + 18
e) -4n + 4
f) -n + 100

Question 3:
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