Special sequences Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Special sequences questions. See exactly how to solve problems on square numbers, recognising sequences, cube numbers, triangular numbers.

square numbersrecognising sequencescube numberstriangular numberscontinuing a sequencearithmetic sequences
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
The square numbers begin 1,4,9,16,…1, 4, 9, 16, \ldots Write down the 6th square number.

Worked solution

  1. Recall what a square number is

    nth square number=n2\text{nth square number} = n^2

    A square number is a whole number multiplied by itself.

  2. Substitute n=6n = 6

    62=6×66^2 = 6 \times 6

    The 6th square number comes from squaring 6.

  3. State the answer

    3636

    A 6 by 6 array of dots holds 36 dots, so the 6th square number is 36.

Answer
3636
Question 2
1 markeasy
The cube numbers begin 1,8,27,…1, 8, 27, \ldots Write down the 4th cube number.

Worked solution

  1. Recall what a cube number is

    nth cube number=n3\text{nth cube number} = n^3

    A cube number is a whole number multiplied by itself three times.

  2. Substitute n=4n = 4

    43=4×4×44^3 = 4 \times 4 \times 4

    The 4th cube number comes from cubing 4.

  3. Work it out

    4×4=16,16×4=644 \times 4 = 16,\quad 16 \times 4 = 64

    The 4th cube number is 64.

Answer
6464
Question 3
1 markeasy
The triangular numbers begin 1,3,6,10,…1, 3, 6, 10, \ldots Write down the 5th triangular number.

Worked solution

  1. See how the triangular numbers grow

    1, 1+2=3, 3+3=6, 6+4=101,\ 1+2=3,\ 3+3=6,\ 6+4=10

    Each triangular number adds one more dot row than the last.

  2. Add the next row

    10+5=1510 + 5 = 15

    The 5th triangle has a bottom row of 5 dots.

  3. State the answer

    T5=15T_5 = 15

    The 5th triangular number is 15 - and 15 dots do form a triangle.

Answer
1515
Question 4
2 markseasy
Here is a sequence of square numbers: 1,4,9,16,251, 4, 9, 16, 25. Write down the next two terms.

Worked solution

  1. Identify the sequence

    1=12, 4=22, 9=32, 16=42, 25=521 = 1^2,\ 4 = 2^2,\ 9 = 3^2,\ 16 = 4^2,\ 25 = 5^2

    These are the square numbers, so the next ones are 6 squared and 7 squared.

  2. Square 6 and square 7

    62=36,72=496^2 = 36,\quad 7^2 = 49

    Carry on squaring the next whole numbers.

  3. State the next two terms

    36, 4936,\ 49

    The differences 3, 5, 7, 9 keep growing by 2, which confirms 25+11=3625 + 11 = 36 and 36+13=4936 + 13 = 49.

Answer
36, 4936,\ 49
Question 5
2 markseasy
Here is a sequence of cube numbers: 1,8,27,641, 8, 27, 64. Write down the next two terms.

Worked solution

  1. Identify the sequence

    1=13, 8=23, 27=33, 64=431 = 1^3,\ 8 = 2^3,\ 27 = 3^3,\ 64 = 4^3

    These are the cube numbers, so the next ones are 5 cubed and 6 cubed.

  2. Cube 5 and cube 6

    53=125,63=2165^3 = 125,\quad 6^3 = 216

    5×5×5=1255 \times 5 \times 5 = 125 and 6×6×6=2166 \times 6 \times 6 = 216.

  3. State the next two terms

    125, 216125,\ 216

    The cube numbers continue 1, 8, 27, 64, 125, 216.

Answer
125, 216125,\ 216

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