High School

3. Fill in the next three numbers in the pattern.

a) 4, 8, 16, ____, ____, ____

b) 5, 10, 20, ____, ____, ____

c) 1600, 800, 400, ____, ____, ____

d) 1, 3, 9, 27, ____, ____, ____

e) 96, 48, 24, ____, ____, ____

f) 3125, 625, 125, ____, ____, ____

Answer :

Let's identify and extend the patterns in each sequence given.

a) The sequence is: 4, 8, 16, ____, ____, ____.
Notice that each number is doubled to get the next number:

  • Start with 4: [tex]4 \times 2 = 8[/tex]
  • Next: [tex]8 \times 2 = 16[/tex]
  • Following this pattern, we get:
    • [tex]16 \times 2 = 32[/tex]
    • [tex]32 \times 2 = 64[/tex]
    • [tex]64 \times 2 = 128[/tex]

So the next numbers are 32, 64, and 128.

b) The sequence is: 5, 10, 20, ____, ____, ____.
Here, too, each number is doubled:

  • Start with 5: [tex]5 \times 2 = 10[/tex]
  • Next: [tex]10 \times 2 = 20[/tex]
  • Continue:
    • [tex]20 \times 2 = 40[/tex]
    • [tex]40 \times 2 = 80[/tex]
    • [tex]80 \times 2 = 160[/tex]

The next numbers are 40, 80, and 160.

c) The sequence is: 1600, 800, 400, ____, ____, ____.
This sequence halves each time:

  • Start with 1600: [tex]\frac{1600}{2} = 800[/tex]
  • Next: [tex]\frac{800}{2} = 400[/tex]
  • Continue:
    • [tex]\frac{400}{2} = 200[/tex]
    • [tex]\frac{200}{2} = 100[/tex]
    • [tex]\frac{100}{2} = 50[/tex]

The next numbers are 200, 100, and 50.

d) The sequence is: 1, 3, 9, 27, ____, ____, ____.
Observe the pattern of multiplying by 3:

  • Start with 1: [tex]1 \times 3 = 3[/tex]
  • Next: [tex]3 \times 3 = 9[/tex]
  • Continue:
    • [tex]9 \times 3 = 27[/tex]
    • [tex]27 \times 3 = 81[/tex]
    • [tex]81 \times 3 = 243[/tex]

The next numbers are 81, 243, and 729.

e) The sequence is: 96, 48, 24, ____, ____, ____.
This pattern involves halving:

  • Start with 96: [tex]\frac{96}{2} = 48[/tex]
  • Next: [tex]\frac{48}{2} = 24[/tex]
  • Continue:
    • [tex]\frac{24}{2} = 12[/tex]
    • [tex]\frac{12}{2} = 6[/tex]
    • [tex]\frac{6}{2} = 3[/tex]

The next numbers are 12, 6, and 3.

f) The sequence is: 3125, 625, 125, ____, ____, ____.
Each term, when divided by 5, produces the next:

  • Start with 3125: [tex]\frac{3125}{5} = 625[/tex]
  • Next: [tex]\frac{625}{5} = 125[/tex]
  • Continue:
    • [tex]\frac{125}{5} = 25[/tex]
    • [tex]\frac{25}{5} = 5[/tex]
    • [tex]\frac{5}{5} = 1[/tex]

The next numbers are 25, 5, and 1.