Math worksheet focusing on sequences and nth terms, featuring exercises to find the first six terms, specific term values, nth term formulas, and solving for unknowns in sequences.

Let's solve each question step-by-step and provide clear explanations.

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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 $
- $ 5(1)=5 $, $ 5(2)=10 $, $ 5(3)=15 $, $ 5(4)=20 $, $ 5(5)=25 $, $ 5(6)=30 $

Answer: $ 5, 10, 15, 20, 25, 30 $

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

Answer: $ 0, 2, 4, 6, 8, 10 $

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

Answer: $ 7, 10, 13, 16, 19, 22 $

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

Answer: $ 2, 5, 8, 11, 14, 17 $

#### f. $ 4n - 3 $
- $ 4(1)-3=1 $, $ 4(2)-3=5 $, $ 4(3)-3=9 $, $ 4(4)-3=13 $, $ 4(5)-3=17 $, $ 4(6)-3=21 $

Answer: $ 1, 5, 9, 13, 17, 21 $

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

Answer: $ 7, 9, 11, 13, 15, 17 $

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

Answer: $ 5, 8, 11, 14, 17, 20 $

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

Answer: $ -2, -1, 0, 1, 2, 3 $

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

Answer: $ -2, 1, 4, 7, 10, 13 $

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

Answer: $ 9, 8, 7, 6, 5, 4 $

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

Answer: $ 17, 14, 11, 8, 5, 2 $

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

Answer: $ 28, 26, 24, 22, 20, 18 $

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

Answer: $ 2, 6, 12, 20, 30, 42 $

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

Answer: $ 0, 3, 8, 15, 24, 35 $

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

Answer: $ 1, 4, 9, 16, 25, 36 $

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

Answer: $ 2, 8, 18, 32, 50, 72 $

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

Answer: $ 4, 16, 36, 64, 100, 144 $

#### s. $ \frac{1}{n} $
- $ \frac{1}{1}=1 $, $ \frac{1}{2}=0.5 $, $ \frac{1}{3} \approx 0.333 $, $ \frac{1}{4}=0.25 $, $ \frac{1}{5}=0.2 $, $ \frac{1}{6} \approx 0.1667 $

Answer: $ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6} $

#### t. $ \frac{n}{2n+1} $
- $ \frac{1}{3} $, $ \frac{2}{5} $, $ \frac{3}{7} $, $ \frac{4}{9} $, $ \frac{5}{11} $, $ \frac{6}{13} $

Answer: $ \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11}, \frac{6}{13} $

#### u. $ \frac{2n-1}{3n+2} $
- $ n=1: \frac{1}{5} $, $ n=2: \frac{3}{8} $, $ n=3: \frac{5}{11} $, $ n=4: \frac{7}{14}=\frac{1}{2} $, $ n=5: \frac{9}{17} $, $ n=6: \frac{11}{20} $

Answer: $ \frac{1}{5}, \frac{3}{8}, \frac{5}{11}, \frac{1}{2}, \frac{9}{17}, \frac{11}{20} $

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

We use $ n = 10 $ and $ n = 50 $ in each formula.

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

Already given as example.

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

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

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

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

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

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

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

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

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

#### k. $ 10 - n $
- $ 10 - 10 = 0 $, $ 10 - 50 = -40 $

#### l. $ 20 - 3n $
- $ 20 - 30 = -10 $, $ 20 - 150 = -130 $

#### m. $ 30 - 2n $
- $ 30 - 20 = 10 $, $ 30 - 100 = -70 $

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

#### o. $ (n-1)(n+1) $
- $ 9×11=99 $, $ 49×51=2499 $

#### p. $ n^2 $
- $ 100 $, $ 2500 $

#### q. $ 2n^2 $
- $ 2(100)=200 $, $ 2(2500)=5000 $

#### r. $ (2n)^2 $
- $ (20)^2 = 400 $, $ (100)^2 = 10000 $

#### s. $ \frac{1}{n} $
- $ \frac{1}{10} = 0.1 $, $ \frac{1}{50} = 0.02 $

#### t. $ \frac{n}{2n+1} $
- $ \frac{10}{21} $, $ \frac{50}{101} $

#### u. $ \frac{2n-1}{3n+2} $
- $ \frac{19}{32} $, $ \frac{99}{152} $

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3. Find the nth term and term 20 for each sequence

We analyze patterns to find $ a_n $, then compute $ a_{20} $.

#### a. $ 1, 3, 5, 7, 9, 11 $ → odd numbers
- $ a_n = 2n - 1 $
- $ a_{20} = 2(20) - 1 = 39 $

#### b. $ 3, 6, 9, 12, 15, 18 $ → multiples of 3
- $ a_n = 3n $
- $ a_{20} = 60 $

#### c. $ 1, 4, 7, 10, 13, 16 $ → common difference 3
- $ a_n = 3n - 2 $
- $ a_{20} = 3(20) - 2 = 58 $

#### d. $ 7, 11, 15, 19, 23, 27 $ → diff = 4
- $ a_n = 4n + 3 $
- $ a_{20} = 4(20) + 3 = 83 $

#### e. $ 4, 7, 10, 13, 16, 19 $ → diff = 3
- $ a_n = 3n + 1 $
- $ a_{20} = 3(20) + 1 = 61 $

#### f. $ 4, 14, 24, 34, 44, 54 $ → diff = 10
- $ a_n = 10n - 6 $
- $ a_{20} = 10(20) - 6 = 194 $

#### g. $ 17, 19, 21, 23, 25, 27 $ → diff = 2
- $ a_n = 2n + 15 $
- $ a_{20} = 2(20) + 15 = 55 $

#### h. $ 2, 6, 10, 14, 18, 22 $ → diff = 4
- $ a_n = 4n - 2 $
- $ a_{20} = 4(20) - 2 = 78 $

#### i. $ 8, 10, 12, 14, 16, 18 $ → diff = 2
- $ a_n = 2n + 6 $
- $ a_{20} = 2(20) + 6 = 46 $

#### j. $ 3, 4, 5, 6, 7, 8 $ → increases by 1
- $ a_n = n + 2 $
- $ a_{20} = 22 $

#### k. $ -4, -1, 2, 5, 8, 11 $ → diff = 3
- $ a_n = 3n - 7 $
- $ a_{20} = 3(20) - 7 = 53 $

#### l. $ 20, 18, 16, 14, 12, 10 $ → diff = -2
- $ a_n = 22 - 2n $
- $ a_{20} = 22 - 40 = -18 $

#### m. $ 7, 4, 1, -2, -5, -8 $ → diff = -3
- $ a_n = 10 - 3n $
- $ a_{20} = 10 - 60 = -50 $

#### n. $ 25, 21, 17, 13, 9, 5 $ → diff = -4
- $ a_n = 29 - 4n $
- $ a_{20} = 29 - 80 = -51 $

#### o. $ \frac{1}{2}, \frac{2}{5}, \frac{3}{8}, \frac{4}{11}, \frac{5}{14} $
- Numerator: $ n $
- Denominator: $ 3n - 1 $
- $ a_n = \frac{n}{3n - 1} $
- $ a_{20} = \frac{20}{59} $

#### p. $ \frac{2}{4}, \frac{4}{7}, \frac{6}{10}, \frac{8}{13}, \frac{10}{16} $
- Num: $ 2n $
- Den: $ 3n + 1 $
- $ a_n = \frac{2n}{3n + 1} $
- $ a_{20} = \frac{40}{61} $

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

Use reverse operations.

#### a. Multiply previous by 2, subtract 3 → 3rd term = 27

Let:
- $ a_3 = 2a_2 - 3 = 27 $
- $ 2a_2 = 30 $ → $ a_2 = 15 $
- $ a_2 = 2a_1 - 3 = 15 $
- $ 2a_1 = 18 $ → $ a_1 = 9 $

Answer: 9

#### b. Multiply by 2, add 4 → 3rd term = 32

- $ a_3 = 2a_2 + 4 = 32 $
- $ 2a_2 = 28 $ → $ a_2 = 14 $
- $ a_2 = 2a_1 + 4 = 14 $
- $ 2a_1 = 10 $ → $ a_1 = 5 $

Answer: 5

#### c. Multiply by 3, subtract 1 → 3rd term = 59

- $ a_3 = 3a_2 - 1 = 59 $
- $ 3a_2 = 60 $ → $ a_2 = 20 $
- $ a_2 = 3a_1 - 1 = 20 $
- $ 3a_1 = 21 $ → $ a_1 = 7 $

Answer: 7

#### d. Add 4 to previous, then multiply by 2 → 3rd term = 36

This means: $ a_n = 2(a_{n-1} + 4) $

So:
- $ a_3 = 2(a_2 + 4) = 36 $
- $ a_2 + 4 = 18 $ → $ a_2 = 14 $
- $ a_2 = 2(a_1 + 4) = 14 $
- $ a_1 + 4 = 7 $ → $ a_1 = 3 $

Answer: 3

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5. Find which term has the value

Solve $ a_n = \text{value} $

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

Answer: 25th term

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

Answer: 18th term

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

Answer: 21st term

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6. Is this number in the sequence? / First term > 100

#### a. Is 37 in $ 4n - 1 $?
Set $ 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

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

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

#### e. First term > 100 in $ 6n - 5 $
Solve $ 6n - 5 > 100 $
→ $ 6n > 105 $ → $ n > 17.5 $ → $ n = 18 $

Check: $ 6(18) - 5 = 108 - 5 = 103 > 100 $

Answer: 18th term

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✔ Final Answers Summary:

1. First 6 terms – see above
2. Term 10 & 50 – see above
3. nth term & term 20 – see above
4. First term
- a. 9
- b. 5
- c. 7
- d. 3

5. Which term?
- a. 25
- b. 18
- c. 21

6.
- a. No
- b. Yes
- c. No
- d. Yes
- e. 18th term

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