Sequences and nth terms worksheet | KS3 maths | Teachit - Free Printable

Let's solve each part of this Sequences and nth Terms worksheet step by step. I'll go through all the questions, providing clear explanations and solutions.

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1. Write the first 6 terms of each sequence

We substitute $ n = 1, 2, 3, 4, 5, 6 $ into each formula.

#### a. $ 2n + 1 $
- $ n=1: 2(1)+1 = 3 $
- $ n=2: 2(2)+1 = 5 $
- $ n=3: 2(3)+1 = 7 $
- $ n=4: 2(4)+1 = 9 $
- $ n=5: 2(5)+1 = 11 $
- $ n=6: 2(6)+1 = 13 $

✔ Answer: 3, 5, 7, 9, 11, 13

#### b. $ 5n $
- $ n=1: 5(1) = 5 $
- $ n=2: 5(2) = 10 $
- $ n=3: 5(3) = 15 $
- $ n=4: 5(4) = 20 $
- $ n=5: 5(5) = 25 $
- $ n=6: 5(6) = 30 $

✔ Answer: 5, 10, 15, 20, 25, 30

#### c. $ 2n - 2 $
- $ n=1: 2(1)-2 = 0 $
- $ n=2: 2(2)-2 = 2 $
- $ n=3: 2(3)-2 = 4 $
- $ n=4: 2(4)-2 = 6 $
- $ n=5: 2(5)-2 = 8 $
- $ n=6: 2(6)-2 = 10 $

✔ Answer: 0, 2, 4, 6, 8, 10

#### d. $ 3n + 4 $
- $ n=1: 3(1)+4 = 7 $
- $ n=2: 3(2)+4 = 10 $
- $ n=3: 3(3)+4 = 13 $
- $ n=4: 3(4)+4 = 16 $
- $ n=5: 3(5)+4 = 19 $
- $ n=6: 3(6)+4 = 22 $

✔ Answer: 7, 10, 13, 16, 19, 22

#### e. $ 3n - 1 $
- $ n=1: 3(1)-1 = 2 $
- $ n=2: 3(2)-1 = 5 $
- $ n=3: 3(3)-1 = 8 $
- $ n=4: 3(4)-1 = 11 $
- $ n=5: 3(5)-1 = 14 $
- $ n=6: 3(6)-1 = 17 $

✔ Answer: 2, 5, 8, 11, 14, 17

#### f. $ 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 $
- $ n=5: 4(5)-3 = 17 $
- $ n=6: 4(6)-3 = 21 $

✔ Answer: 1, 5, 9, 13, 17, 21

#### g. $ 2n + 5 $
- $ n=1: 2(1)+5 = 7 $
- $ n=2: 2(2)+5 = 9 $
- $ n=3: 2(3)+5 = 11 $
- $ n=4: 2(4)+5 = 13 $
- $ n=5: 2(5)+5 = 15 $
- $ n=6: 2(6)+5 = 17 $

✔ Answer: 7, 9, 11, 13, 15, 17

#### h. $ 3n + 2 $
- $ n=1: 3(1)+2 = 5 $
- $ n=2: 3(2)+2 = 8 $
- $ n=3: 3(3)+2 = 11 $
- $ n=4: 3(4)+2 = 14 $
- $ n=5: 3(5)+2 = 17 $
- $ n=6: 3(6)+2 = 20 $

✔ Answer: 5, 8, 11, 14, 17, 20

#### i. $ n - 3 $
- $ n=1: 1 - 3 = -2 $
- $ n=2: 2 - 3 = -1 $
- $ n=3: 3 - 3 = 0 $
- $ n=4: 4 - 3 = 1 $
- $ n=5: 5 - 3 = 2 $
- $ n=6: 6 - 3 = 3 $

✔ Answer: -2, -1, 0, 1, 2, 3

#### j. $ 3n - 5 $
- $ n=1: 3(1)-5 = -2 $
- $ n=2: 3(2)-5 = 1 $
- $ n=3: 3(3)-5 = 4 $
- $ n=4: 3(4)-5 = 7 $
- $ n=5: 3(5)-5 = 10 $
- $ n=6: 3(6)-5 = 13 $

✔ Answer: -2, 1, 4, 7, 10, 13

#### k. $ 10 - n $
- $ n=1: 10 - 1 = 9 $
- $ n=2: 10 - 2 = 8 $
- $ n=3: 10 - 3 = 7 $
- $ n=4: 10 - 4 = 6 $
- $ n=5: 10 - 5 = 5 $
- $ n=6: 10 - 6 = 4 $

✔ Answer: 9, 8, 7, 6, 5, 4

#### l. $ 20 - 3n $
- $ n=1: 20 - 3(1) = 17 $
- $ n=2: 20 - 3(2) = 14 $
- $ n=3: 20 - 3(3) = 11 $
- $ n=4: 20 - 3(4) = 8 $
- $ n=5: 20 - 3(5) = 5 $
- $ n=6: 20 - 3(6) = 2 $

✔ Answer: 17, 14, 11, 8, 5, 2

#### m. $ 30 - 2n $
- $ n=1: 30 - 2(1) = 28 $
- $ n=2: 30 - 2(2) = 26 $
- $ n=3: 30 - 2(3) = 24 $
- $ n=4: 30 - 2(4) = 22 $
- $ n=5: 30 - 2(5) = 20 $
- $ n=6: 30 - 2(6) = 18 $

✔ Answer: 28, 26, 24, 22, 20, 18

#### n. $ n(n+1) $
- $ n=1: 1(1+1) = 1×2 = 2 $
- $ n=2: 2(2+1) = 2×3 = 6 $
- $ n=3: 3(3+1) = 3×4 = 12 $
- $ n=4: 4(4+1) = 4×5 = 20 $
- $ n=5: 5(5+1) = 5×6 = 30 $
- $ n=6: 6(6+1) = 6×7 = 42 $

✔ Answer: 2, 6, 12, 20, 30, 42

#### o. $ (n-1)(n+1) $
Note: $ (n-1)(n+1) = n^2 - 1 $
- $ n=1: (0)(2) = 0 $
- $ n=2: (1)(3) = 3 $
- $ n=3: (2)(4) = 8 $
- $ n=4: (3)(5) = 15 $
- $ n=5: (4)(6) = 24 $
- $ n=6: (5)(7) = 35 $

✔ Answer: 0, 3, 8, 15, 24, 35

#### p. $ n^2 $
- $ n=1: 1^2 = 1 $
- $ n=2: 2^2 = 4 $
- $ n=3: 3^2 = 9 $
- $ n=4: 4^2 = 16 $
- $ n=5: 5^2 = 25 $
- $ n=6: 6^2 = 36 $

✔ Answer: 1, 4, 9, 16, 25, 36

#### q. $ 2n^2 $
- $ n=1: 2(1)^2 = 2 $
- $ n=2: 2(4) = 8 $
- $ n=3: 2(9) = 18 $
- $ n=4: 2(16) = 32 $
- $ n=5: 2(25) = 50 $
- $ n=6: 2(36) = 72 $

✔ Answer: 2, 8, 18, 32, 50, 72

#### r. $ (2n)^2 $
- $ n=1: (2)^2 = 4 $
- $ n=2: (4)^2 = 16 $
- $ n=3: (6)^2 = 36 $
- $ n=4: (8)^2 = 64 $
- $ n=5: (10)^2 = 100 $
- $ n=6: (12)^2 = 144 $

✔ Answer: 4, 16, 36, 64, 100, 144

#### s. $ \frac{1}{n} $
- $ n=1: 1/1 = 1 $
- $ n=2: 1/2 $
- $ n=3: 1/3 $
- $ n=4: 1/4 $
- $ n=5: 1/5 $
- $ n=6: 1/6 $

✔ Answer: 1, ½, ⅓, ¼, ⅕, ⅙

#### t. $ \frac{n}{2n+1} $
- $ n=1: 1/(2+1) = 1/3 $
- $ n=2: 2/(4+1) = 2/5 $
- $ n=3: 3/(6+1) = 3/7 $
- $ n=4: 4/(8+1) = 4/9 $
- $ n=5: 5/(10+1) = 5/11 $
- $ n=6: 6/(12+1) = 6/13 $

✔ Answer: 1/3, 2/5, 3/7, 4/9, 5/11, 6/13

#### u. $ \frac{2n-1}{3n+2} $
- $ n=1: (2-1)/(3+2) = 1/5 $
- $ n=2: (4-1)/(6+2) = 3/8 $
- $ n=3: (6-1)/(9+2) = 5/11 $
- $ n=4: (8-1)/(12+2) = 7/14 = 1/2 $
- $ n=5: (10-1)/(15+2) = 9/17 $
- $ n=6: (12-1)/(18+2) = 11/20 $

✔ Answer: 1/5, 3/8, 5/11, 1/2, 9/17, 11/20

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2. Find term 10 and term 50 for each sequence in Q1

We plug in $ n = 10 $ and $ n = 50 $ into each formula.

#### a. $ 2n + 1 $
- Term 10: $ 2(10)+1 = 21 $
- Term 50: $ 2(50)+1 = 101 $

✔ Already done as example.

#### b. $ 5n $
- Term 10: $ 5(10) = 50 $
- Term 50: $ 5(50) = 250 $

#### c. $ 2n - 2 $
- Term 10: $ 2(10)-2 = 18 $
- Term 50: $ 2(50)-2 = 98 $

#### d. $ 3n + 4 $
- Term 10: $ 3(10)+4 = 34 $
- Term 50: $ 3(50)+4 = 154 $

#### e. $ 3n - 1 $
- Term 10: $ 3(10)-1 = 29 $
- Term 50: $ 3(50)-1 = 149 $

#### f. $ 4n - 3 $
- Term 10: $ 4(10)-3 = 37 $
- Term 50: $ 4(50)-3 = 197 $

#### g. $ 2n + 5 $
- Term 10: $ 2(10)+5 = 25 $
- Term 50: $ 2(50)+5 = 105 $

#### h. $ 3n + 2 $
- Term 10: $ 3(10)+2 = 32 $
- Term 50: $ 3(50)+2 = 152 $

#### i. $ n - 3 $
- Term 10: $ 10 - 3 = 7 $
- Term 50: $ 50 - 3 = 47 $

#### j. $ 3n - 5 $
- Term 10: $ 3(10)-5 = 25 $
- Term 50: $ 3(50)-5 = 145 $

#### k. $ 10 - n $
- Term 10: $ 10 - 10 = 0 $
- Term 50: $ 10 - 50 = -40 $

#### l. $ 20 - 3n $
- Term 10: $ 20 - 30 = -10 $
- Term 50: $ 20 - 150 = -130 $

#### m. $ 30 - 2n $
- Term 10: $ 30 - 20 = 10 $
- Term 50: $ 30 - 100 = -70 $

#### n. $ n(n+1) $
- Term 10: $ 10×11 = 110 $
- Term 50: $ 50×51 = 2550 $

#### o. $ (n-1)(n+1) $
- Term 10: $ 9×11 = 99 $
- Term 50: $ 49×51 = 2499 $

#### p. $ n^2 $
- Term 10: $ 100 $
- Term 50: $ 2500 $

#### q. $ 2n^2 $
- Term 10: $ 2×100 = 200 $
- Term 50: $ 2×2500 = 5000 $

#### r. $ (2n)^2 $
- Term 10: $ (20)^2 = 400 $
- Term 50: $ (100)^2 = 10000 $

#### s. $ \frac{1}{n} $
- Term 10: $ 1/10 $
- Term 50: $ 1/50 $

#### t. $ \frac{n}{2n+1} $
- Term 10: $ 10/(20+1) = 10/21 $
- Term 50: $ 50/(100+1) = 50/101 $

#### u. $ \frac{2n-1}{3n+2} $
- Term 10: $ (20-1)/(30+2) = 19/32 $
- Term 50: $ (100-1)/(150+2) = 99/152 $

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3. Find the nth term and value of term 20

We identify patterns and find general formulas.

#### a. 1, 3, 5, 7, 9, 11
→ Arithmetic sequence with common difference 2.
First term $ a = 1 $, $ d = 2 $
→ $ a_n = 1 + (n-1)×2 = 2n - 1 $

Term 20: $ 2(20) - 1 = 39 $

✔ nth term: $ 2n - 1 $, Term 20: 39

#### b. 3, 6, 9, 12, 15, 18
→ $ d = 3 $, $ a = 3 $
→ $ a_n = 3n $

Term 20: $ 3×20 = 60 $

✔ nth term: $ 3n $, Term 20: 60

#### c. 1, 4, 7, 10, 13, 16
→ $ d = 3 $, $ a = 1 $
→ $ a_n = 1 + (n-1)×3 = 3n - 2 $

Term 20: $ 3(20) - 2 = 58 $

✔ nth term: $ 3n - 2 $, Term 20: 58

#### d. 7, 11, 15, 19, 23, 27
→ $ d = 4 $, $ a = 7 $
→ $ a_n = 7 + (n-1)×4 = 4n + 3 $

Check: $ n=1: 4+3=7 $ ✔

Term 20: $ 4(20)+3 = 83 $

✔ nth term: $ 4n + 3 $, Term 20: 83

#### e. 4, 7, 10, 13, 16, 19
→ $ d = 3 $, $ a = 4 $
→ $ a_n = 4 + (n-1)×3 = 3n + 1 $

Term 20: $ 3(20)+1 = 61 $

✔ nth term: $ 3n + 1 $, Term 20: 61

#### f. 4, 14, 24, 34, 44, 54
→ $ d = 10 $, $ a = 4 $
→ $ a_n = 4 + (n-1)×10 = 10n - 6 $

Check: $ n=1: 10-6=4 $ ✔

Term 20: $ 10(20) - 6 = 194 $

✔ nth term: $ 10n - 6 $, Term 20: 194

#### g. 17, 19, 21, 23, 25, 27
→ $ d = 2 $, $ a = 17 $
→ $ a_n = 17 + (n-1)×2 = 2n + 15 $

Check: $ n=1: 2+15=17 $ ✔

Term 20: $ 2(20)+15 = 55 $

✔ nth term: $ 2n + 15 $, Term 20: 55

#### h. 2, 6, 10, 14, 18, 22
→ $ d = 4 $, $ a = 2 $
→ $ a_n = 2 + (n-1)×4 = 4n - 2 $

Term 20: $ 4(20) - 2 = 78 $

✔ nth term: $ 4n - 2 $, Term 20: 78

#### i. 8, 10, 12, 14, 16, 18
→ $ d = 2 $, $ a = 8 $
→ $ a_n = 8 + (n-1)×2 = 2n + 6 $

Term 20: $ 2(20)+6 = 46 $

✔ nth term: $ 2n + 6 $, Term 20: 46

#### j. 3, 4, 5, 6, 7, 8
→ $ d = 1 $, $ a = 3 $
→ $ a_n = 3 + (n-1)×1 = n + 2 $

Term 20: $ 20 + 2 = 22 $

✔ nth term: $ n + 2 $, Term 20: 22

#### k. -4, -1, 2, 5, 8, 11
→ $ d = 3 $, $ a = -4 $
→ $ a_n = -4 + (n-1)×3 = 3n - 7 $

Check: $ n=1: 3-7=-4 $ ✔

Term 20: $ 3(20)-7 = 53 $

✔ nth term: $ 3n - 7 $, Term 20: 53

#### l. 20, 18, 16, 14, 12, 10
→ $ d = -2 $, $ a = 20 $
→ $ a_n = 20 + (n-1)(-2) = -2n + 22 $

Term 20: $ -2(20)+22 = -18 $

✔ nth term: $ -2n + 22 $, Term 20: -18

#### m. 7, 4, 1, -2, -5, -8
→ $ d = -3 $, $ a = 7 $
→ $ a_n = 7 + (n-1)(-3) = -3n + 10 $

Term 20: $ -3(20)+10 = -50 $

✔ nth term: $ -3n + 10 $, Term 20: -50

#### n. 25, 21, 17, 13, 9, 5
→ $ d = -4 $, $ a = 25 $
→ $ a_n = 25 + (n-1)(-4) = -4n + 29 $

Term 20: $ -4(20)+29 = -51 $

✔ nth term: $ -4n + 29 $, Term 20: -51

#### o. $ \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, \frac{5}{14} $
Numerator: $ n $
Denominator: 2, 5, 8, 11, 14 → $ 3n - 1 $
So: $ a_n = \frac{n}{3n - 1} $

Term 20: $ \frac{20}{3(20)-1} = \frac{20}{59} $

✔ nth term: $ \frac{n}{3n - 1} $, Term 20: $ \frac{20}{59} $

#### p. $ \frac{2}{4}, \frac{4}{7}, \frac{6}{10}, \frac{8}{13}, \frac{10}{16} $
Numerator: $ 2n $
Denominator: 4, 7, 10, 13, 16 → $ 3n + 1 $
So: $ a_n = \frac{2n}{3n + 1} $

Term 20: $ \frac{40}{61} $

✔ nth term: $ \frac{2n}{3n + 1} $, Term 20: $ \frac{40}{61} $

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4. Work out the first term given rule and third term

We work backwards from the third term using the rule.

#### a. Multiply previous term by 2 then subtract 3
Third term = 27

Let second term be $ x $. Then:
$ 2x - 3 = 27 $ → $ 2x = 30 $ → $ x = 15 $

Now let first term be $ y $:
$ 2y - 3 = 15 $ → $ 2y = 18 $ → $ y = 9 $

✔ First term: 9

#### b. Multiply previous term by 2 then add 4
Third term = 32

Let second term = $ x $:
$ 2x + 4 = 32 $ → $ 2x = 28 $ → $ x = 14 $

Let first term = $ y $:
$ 2y + 4 = 14 $ → $ 2y = 10 $ → $ y = 5 $

✔ First term: 5

#### c. Multiply previous term by 3 then subtract 1
Third term = 59

Let second term = $ x $:
$ 3x - 1 = 59 $ → $ 3x = 60 $ → $ x = 20 $

Let first term = $ y $:
$ 3y - 1 = 20 $ → $ 3y = 21 $ → $ y = 7 $

✔ First term: 7

#### d. Add 4 to previous term then multiply by 2
Third term = 36

Let second term = $ x $:
$ 2(x + 4) = 36 $ → $ x + 4 = 18 $ → $ x = 14 $

Let first term = $ y $:
$ 2(y + 4) = 14 $ → $ y + 4 = 7 $ → $ y = 3 $

✔ First term: 3

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5. Find which term has the given value

Set $ a_n = \text{value} $, solve for $ n $

#### a. $ 3n + 1 = 76 $
→ $ 3n = 75 $ → $ n = 25 $

✔ Term 25

#### b. $ 2n - 5 = 31 $
→ $ 2n = 36 $ → $ n = 18 $

✔ Term 18

#### c. $ 4n - 2 = 82 $
→ $ 4n = 84 $ → $ n = 21 $

✔ Term 21

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6. Check if number is in sequence or find first term > 100

#### a. Is 37 in $ 4n - 1 $?
Solve: $ 4n - 1 = 37 $ → $ 4n = 38 $ → $ n = 9.5 $ → not integer

✘ No

#### b. Is 71 in $ 2n + 3 $?
$ 2n + 3 = 71 $ → $ 2n = 68 $ → $ n = 34 $ → yes

✔ Yes, it is the 34th term

#### c. Is 60 in $ 5n + 4 $?
$ 5n + 4 = 60 $ → $ 5n = 56 $ → $ n = 11.2 $ → no

✘ No

#### d. Is 40 in $ 3n - 5 $?
$ 3n - 5 = 40 $ → $ 3n = 45 $ → $ n = 15 $ → yes

✔ Yes, term 15

#### e. First term > 100 in $ 6n - 5 $
Solve: $ 6n - 5 > 100 $ → $ 6n > 105 $ → $ n > 17.5 $

So smallest integer $ n = 18 $

Check: $ 6(18) - 5 = 108 - 5 = 103 $

✔ First term greater than 100 is 103, at $ n = 18 $

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✔ Final Answers Summary

Due to length, I've provided full working above. Here are key answers:

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Q1: First 6 terms — solved for all.

Q2: Term 10 & 50 — solved.

Q3:
a. $ 2n-1 $, 39
b. $ 3n $, 60
c. $ 3n-2 $, 58
d. $ 4n+3 $, 83
e. $ 3n+1 $, 61
f. $ 10n-6 $, 194
g. $ 2n+15 $, 55
h. $ 4n-2 $, 78
i. $ 2n+6 $, 46
j. $ n+2 $, 22
k. $ 3n-7 $, 53
l. $ -2n+22 $, -18
m. $ -3n+10 $, -50
n. $ -4n+29 $, -51
o. $ \frac{n}{3n-1} $, $ \frac{20}{59} $
p. $ \frac{2n}{3n+1} $, $ \frac{40}{61} $

Q4:
a. 9
b. 5
c. 7
d. 3

Q5:
a. 25
b. 18
c. 21

Q6:
a. No
b. Yes (term 34)
c. No
d. Yes (term 15)
e. 103 (at $ n=18 $)

Let me know if you'd like these compiled into a clean answer sheet!