High School

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1. Write down the next three numbers in each of the sequences below. Also, explain in writing, in each case, how you figured out what the numbers should be.

a) Sequence A: 2; 5; 8; 11; 14; 17; 20; 23;

b) Sequence B: 4; 5; 8; 13; 20; 29; 40;

c) Sequence C: 1; 2; 4; 8; 16; 32; 64;

d) Sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19;

e) Sequence E: 4; 5; 7; 10; 14; 19; 25; 32; 40;

f) Sequence F: 2; 6; 18; 54; 162; 486;

g) Sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33;

Answer :

Let's look at each sequence one by one to find the next three numbers and understand the pattern.

a) Sequence A: 2; 5; 8; 11; 14; 17; 20; 23

The pattern here is that each number increases by 3. Starting from 23:

  • 23 + 3 = 26
  • 26 + 3 = 29
  • 29 + 3 = 32

So, the next three numbers are: 26, 29, 32.

b) Sequence B: 4; 5; 8; 13; 20; 29; 40

Let's calculate the differences between consecutive numbers to find the pattern:

  • 5 - 4 = 1
  • 8 - 5 = 3
  • 13 - 8 = 5
  • 20 - 13 = 7
  • 29 - 20 = 9
  • 40 - 29 = 11

The differences (1, 3, 5, 7, 9, 11) form an increasing sequence of odd numbers, adding 2 each time. The next differences will be 13, 15, and 17:

  • 40 + 13 = 53
  • 53 + 15 = 68
  • 68 + 17 = 85

So, the next three numbers are: 53, 68, 85.

c) Sequence C: 1; 2; 4; 8; 16; 32; 64

In this sequence, each number is obtained by multiplying the previous number by 2:

  • 64 \times 2 = 128
  • 128 \times 2 = 256
  • 256 \times 2 = 512

So, the next three numbers are: 128, 256, 512.

d) Sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19

Each number in this sequence increases by 2:

  • 19 + 2 = 21
  • 21 + 2 = 23
  • 23 + 2 = 25

So, the next three numbers are: 21, 23, 25.

e) Sequence E: 4; 5; 7; 10; 14; 19; 25; 32; 40

The differences are:

  • 5 - 4 = 1
  • 7 - 5 = 2
  • 10 - 7 = 3
  • 14 - 10 = 4
  • 19 - 14 = 5
  • 25 - 19 = 6
  • 32 - 25 = 7
  • 40 - 32 = 8

The differences are consecutive whole numbers increasing by 1. So, the next differences will be 9, 10, and 11:

  • 40 + 9 = 49
  • 49 + 10 = 59
  • 59 + 11 = 70

So, the next three numbers are: 49, 59, 70.

f) Sequence F: 2; 6; 18; 54; 162; 486

In this sequence, each number is obtained by multiplying the previous number by 3:

  • 486 \times 3 = 1458
  • 1458 \times 3 = 4374
  • 4374 \times 3 = 13122

So, the next three numbers are: 1458, 4374, 13122.

g) Sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33

Every number in this sequence increases by 4:

  • 33 + 4 = 37
  • 37 + 4 = 41
  • 41 + 4 = 45

So, the next three numbers are: 37, 41, 45.

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