Skip to main content
Ecolint Campus des NationsMathematics
← Review packs

Pack A — Fluency

Ecolint Campus des NationsMathematics
Year 8 · 8.6 Patterns

Pack A · Fluency

Name: _________________________________
Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10
  1. 1.
    Find the next two terms of the sequence 5, 8, 11, 14, … and describe the rule.
     
  2. 2.
    The nn-th term of a sequence is Tn=4n+1T_n = 4n + 1. Find T10T_10.
     
  3. 3.
    A pattern of dots has 4 dots in the 1st, 7 dots in the 2nd, 10 dots in the 3rd. (a) How many dots in the 4th? (b) Describe the rule.
     
  4. 4.
    Find the next term of: 1, 4, 9, …
     
  5. 5.
    Continue: −3,−1,1,3,…-3, -1, 1, 3, \ldots
     
  6. 6.
    For the sequence with formula Tn=2n+5T_n = 2n + 5, find the first three terms.
     
  7. 7.
    A pattern of stars: 1st has 5, 2nd has 8, 3rd has 11. Find a rule.
     
  8. 8.
    Find the common difference of the sequence 12, 8, 4, 0, −4, …
     
  9. 9.
    Verify whether 47 is a term of the sequence Tn=3n+2T_n = 3n + 2.
     
  10. 10.
    Verify whether 40 is a term of the sequence Tn=3n+2T_n = 3n + 2.
     
SilverQuestions 11–20
  1. 11.
    Find a formula for the nn-th term of the sequence 4, 7, 10, 13, …
     
  2. 12.
    Find the 20th term of Tn=5n−2T_n = 5n - 2.
     
  3. 13.
    A pattern of matchsticks: each new triangle adds 2 sticks. The first uses 3 sticks. Find a formula for TnT_n.
     
  4. 14.
    A sequence has first term 10 and common difference −3-3. Write a formula for TnT_n.
     
  5. 15.
    Find nn when Tn=71T_n = 71 in the sequence Tn=4n−5T_n = 4n - 5.
     
  6. 16.
    A linear sequence has T3=11T_3 = 11 and T7=27T_7 = 27. (a) Find dd. (b) Find T1T_1. (c) Write TnT_n.
     
  7. 17.
    A square pattern grows: 1st has 1 dot, 2nd has 4, 3rd has 9. Write a formula.
     
  8. 18.
    A sequence: 1,3,6,10,15,…1, 3, 6, 10, 15, \ldots (triangular numbers). Find T6T_6 and T7T_7.
     
  9. 19.
    A sequence of stair-step shapes: 1st uses 1 cube, 2nd uses 3, 3rd uses 6, 4th uses 10. Find the rule.
     
  10. 20.
    The Fibonacci sequence is 1, 1, 2, 3, 5, 8, … State the recursion and find T8T_8.
     
GoldQuestions 21–30
  1. 21.
    A row of conference tables seats people. The pattern is M=6n+2M = 6n + 2. (a) How many people sit at 8 tables? (b) How many tables are needed to seat at least 100 people?
     
  2. 22.
    In Mr. Packer's arrangement, aa people sit on each side, bb at each end. Show P=2an+2bP = 2an + 2b and find PP when a=5a = 5, b=3b = 3, n=4n = 4.
     
  3. 23.
    Patterns of matchsticks: 1st 3 sticks, 2nd 5, 3rd 7. (a) Formula for TnT_n. (b) Which pattern uses 51 sticks?
     
  4. 24.
    Find a formula and the 50th term of: 7, 11, 15, 19, …
     
  5. 25.
    A growing pattern of squares uses 4 sticks for the first square, then adds 3 sticks per extra square in a row. (a) Formula for TnT_n. (b) How many squares can you make from 100 sticks? (c) How many sticks left over?
     
  6. 26.
    Find a formula for TnT_n in the sequence −1,2,5,8,11,…-1, 2, 5, 8, 11, \ldots
     
  7. 27.
    The 4th term of a linear sequence is 17 and the 10th term is 41. Find a formula for TnT_n.
     
  8. 28.
    The sequence 5,8,13,20,29,…5, 8, 13, 20, 29, \ldots has second differences 2. Find a formula for TnT_n.
     
  9. 29.
    Find the sum of the first 10 terms of Tn=2n+1T_n = 2n + 1 (odd numbers starting from 3).
     
  10. 30.
    For the pattern Tn=n2−nT_n = n^2 - n, find T1,T2,T5,T10T_1, T_2, T_5, T_{10}.
     
PlatinumQuestions 31–40
  1. 31.
    The sequence of perfect squares is 1,4,9,16,25,…1, 4, 9, 16, 25, \ldots (a) Formula for TnT_n. (b) Differences between consecutive terms. (c) Use this to find T20T_{20} without squaring.
     
  2. 32.
    The 5th term of a linear sequence is 18 and the 12th term is 53. Find a formula, and find the smallest nn for which Tn>100T_n > 100.
     
  3. 33.
    For the sequence 1,5,12,22,35,…1, 5, 12, 22, 35, \ldots (pentagonal numbers, formula Pn=n(3n−1)2P_n = \tfrac{n(3n-1)}{2}): verify the formula for n=1,2,3n = 1, 2, 3, and find P10P_{10}.
     
  4. 34.
    Find the sum of all multiples of 3 between 1 and 100 (inclusive).
     
  5. 35.
    A linear sequence has first term aa and common difference dd. (a) Express T5T_5 in terms of aa and dd. (b) If T5=23T_5 = 23 and T12=65T_{12} = 65, find aa and dd.
     
  6. 36.
    Two arithmetic sequences A: 5, 8, 11, … and B: 2, 7, 12, … . (a) Find formulas. (b) For which nn are the nn-th terms equal?
     
  7. 37.
    A figurate-number pattern: the nn-th hexagonal number is Hn=n(2n−1)H_n = n(2n-1). Find H6H_6 and H10H_{10}.
     
  8. 38.
    A geometric sequence has first term 3 and common ratio 2. (a) Find the first 5 terms. (b) Find a formula TnT_n. (c) Find T10T_{10}.
     
  9. 39.
    A staircase pattern uses 1, 4, 9, 16, … cubes per layer. (a) State the formula for the nn-th layer. (b) Find the total number of cubes in the first 5 layers. (c) Find a formula for the total Sn=12+22+…+n2S_n = 1^2 + 2^2 + \ldots + n^2 if you can.
     
  10. 40.
    A sequence has Tn=2n2−nT_n = 2n^2 - n. (a) Find T1,T2,T3T_1, T_2, T_3. (b) Show that the second differences are constant and find their value. (c) Is 119 a term of the sequence?