High School

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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