Describing Patterns Algebraically — Solutions

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1. Common difference and next 3 terms

   1. 2, 5, 8, 11… (d=3): 14, 17, 20
   2. 10, 7, 4, 1… (d=−3): −2, −5, −8
   3. 3, 8, 13, 18… (d=5): 23, 28, 33
   4. 20, 16, 12, 8… (d=−4): 4, 0, −4
   5. 1, 1.5, 2, 2.5… (d=0.5): 3, 3.5, 4
   6. −3, 0, 3, 6… (d=3): 9, 12, 15
   7. 100, 93, 86, 79… (d=−7): 72, 65, 58
   8. 4, 4.5, 5, 5.5… (d=0.5): 6, 6.5, 7

2. nth term formula

   1. 3, 7, 11, 15…: T_n = 4n − 1
   2. 5, 8, 11, 14…: T_n = 3n + 2
   3. 10, 8, 6, 4…: T_n = −2n + 12
   4. 1, 4, 7, 10…: T_n = 3n − 2
   5. 6, 11, 16, 21…: T_n = 5n + 1
   6. 20, 17, 14, 11…: T_n = −3n + 23
   7. 2, 2.5, 3, 3.5…: T_n = 0.5n + 1.5
   8. −1, 2, 5, 8…: T_n = 3n − 4

3. Find specific terms

   1. T_n = 3n+1: T₂₀=61, T₅₀=151, T₁₀₀=301
   2. T_n = 5n−2: T₂₀=98, T₅₀=248, T₁₀₀=498
   3. T_n = −2n+50: T₁₀=30, T₂₀=10, T₂₅=0
   4. 4, 9, 14, 19… (T_n=5n−1): T₂₀=99, T₅₀=249, T₁₀₀=499
   5. 7, 11, 15, 19… (T_n=4n+3): T₂₀=83, T₅₀=203, T₁₀₀=403
   6. 200, 195, 190, 185… (T_n=−5n+205): T₂₀=105, T₄₀=5, T₅₀=−45
   7. T_n = 2n+0.5: T₁₀=20.5, T₅₀=100.5, T₁₀₀=200.5
   8. −5, −1, 3, 7… (T_n=4n−9): T₁₅=51, T₃₀=111, T₅₀=191

4. Is a value in the sequence?

   1. 5, 8, 11, 14… (T_n=3n+2), is 47 a term?: 3n+2=47 → n=15. Yes, T₁₅=47.
   2. 3, 7, 11, 15… (T_n=4n−1), is 100 a term?: 4n−1=100 → n=25.25. Not a whole number, so 100 is NOT in the sequence.
   3. 4, 9, 14, 19… (T_n=5n−1), is 85 a term?: 5n−1=85 → n=17.2. Not a whole number. No.
   4. 6, 11, 16, 21… (T_n=5n+1), is 200 a term?: 5n+1=200 → n=39.8. Not a whole number. No.
   5. 2, 6, 10, 14… (T_n=4n−2), is 55 a term?: 4n−2=55 → n=57/4=14.25. No.
   6. 10, 7, 4, 1… (T_n=−3n+13), is −40 a term?: −3n+13=−40 → n=53/3. Not a whole number. No.
   7. 7, 14, 21, 28… (T_n=7n), is 1000 a term?: 7n=1000 → n=142.86. No, 1000 is not divisible by 7.
   8. 8, 15, 22, 29… (T_n=7n+1), is 72 a term?: 7n+1=72 → n=71/7. Not a whole number. No.

5. Geometric shape patterns

   1. Matchstick squares:
      1. 4, 7, 10, 13, 16
      2. T_n = 3n + 1
      3. T₂₀ = 61 sticks
   2. Matchstick triangles:
      1. T_n = 2n + 1
      2. 2n+1=31 → n=15. 15 triangles.
   3. L-shape dots:
      1. T_n = 2n + 1
      2. 2n+1=25 → n=12. Stage 12.
   4. Plus-shape tiles:
      1. T_n = 4n + 1
      2. 4n+1=49 → n=12. Stage 12.
   5. Hexagon sticks:
      1. T_n = 5n + 1
      2. T₁₀ = 51 sticks
   6. Square spiral:
      1. T_n = 2n − 1
      2. 2n−1=100 → n=50.5. No — this is not a whole number, so 100 squares is not possible. The sequence produces only odd numbers.

6. Problem solving

   1. Cinema seats:
      1. T_n = 4n + 16
      2. T₁₅ = 76 seats
      3. Sum = T₁+…+T₂₀ = 20×20 + (4)(20×21/2) = 400 + 840 = 1240 seats
   2. Pool filling:
      1. T_n = 600n + 200
      2. 600n+200=5000 → n=8 hours
      3. T₁₂=7400 L. No, still 600 L short of 8000 L.
   3. Two sequences:
      1. A: T_n=4n−1; B: T_n=3n+2
      2. Set equal: 4n−1=3m+2 for whole numbers n,m ≤20. Common values: 11 (A: n=3, B: m=3); 23 (A: n=6, B: m=7); 35, 47, 59, 71… every 12th value.
      3. Yes, they share a term every 12 values (LCM of common differences 4 and 3). The shared terms form the sequence 11, 23, 35… with rule T_n=12n−1.
   4. Can display:
      1. Row n from bottom has (11−n) cans, so T_n = 11−n (or: top row is row 10 with 1 can, bottom is row 1 with 10 cans)
      2. 11−n=6 → n=5. Row 5.
      3. Total = 1+2+…+10 = 55 cans

7. Linear vs non-linear patterns

   1. 1, 4, 9, 16, 25…: Non-linear — differences are 3, 5, 7, 9 (not constant). These are square numbers.
   2. 3, 5, 7, 9, 11…: Linear (d=2). T_n = 2n + 1.
   3. 2, 4, 8, 16, 32…: Non-linear — each term doubles (ratio is constant, not difference). This is an exponential (geometric) sequence.
   4. 10, 7, 4, 1, −2…: Linear (d=−3). T_n = −3n + 13.
   5. 0, 3, 8, 15, 24…: Non-linear — differences are 3, 5, 7, 9 (increasing). The second differences are constant (=2), making this a quadratic pattern.
   6. 100, 97, 94, 91…: Linear (d=−3). T_n = −3n + 103.

8. Two company patterns — when do they meet?

   1. Company A rule: a=12, d=5. T_n = 5n + 7.
   2. Company B rule: a=32, d=2. T_n = 2n + 30.
   3. When equal: 5n + 7 = 2n + 30 → 3n = 23 → n = 23/3, which is not a whole number. The sequences never produce the exact same value in the same week. (Company A overtakes Company B between week 7 and 8: T₇: A=42, B=44; T₈: A=47, B=46.)
   4. Total sales after 10 weeks: A: T₁…T₁₀ = 12+17+22+27+32+37+42+47+52+57 = 345. B: 32+34+36+38+40+42+44+46+48+50 = 410. Company B has more total sales after 10 weeks.

9. Fencing panels

   1. Panels table: 1 paddock: 4; 2: 7; 3: 10; 4: 13; 5: 16
   2. nth term rule: d=3, a=4. T_n = 3n + 1.
   3. 25 paddocks: T₂₅ = 3(25) + 1 = 76 panels.
   4. Maximum paddocks with 76 panels: 3n + 1 = 76 → 3n = 75 → n = 25 paddocks exactly.

10. Working backwards — reverse sequence problems

    1. T_n = 7n − 3, value 375: 7n − 3 = 375 → 7n = 378 → n = 54. T₅₄ = 375.
    2. Sequence 8, 14, 20… first exceeds 200: T_n = 6n + 2. 6n + 2 > 200 → n > 198/6 = 33. So n = 34. T₃₄ = 206.
    3. Sequence 150, 146, 142…, last positive term: T_n = −4n + 154. Set −4n + 154 > 0 → n < 38.5. Last positive term is T₃₈ = −4(38) + 154 = 2.
    4. Finding rules and common term: Sequence P: T₁=5, T₄=17. d = (17−5)/3 = 4. T_n = 4n + 1. Sequence Q: T₂=10, T₅=19. d = (19−10)/3 = 3. T_n = 3n + 4. Set equal: 4n + 1 = 3m + 4 for whole numbers. Check values: P: 5,9,13,17,21,25,29,33…; Q: 7,10,13,16,19… First common value is 13 (P: n=3, Q: m=3).