Question 4

We will find the next three numbers for each sequence and describe the rule.

Step 1 — All 1's

This sequence has only the number 1. Every number in this sequence is the same.

The given sequence is 1, 1, 1, 1.

We need to find the next three numbers.

$1$

$1$

$1$

$\boxed{\text{The next three numbers are 1, 1, 1. The rule is that every number is 1.}}$

Step 2 — Counting numbers

This sequence lists numbers in order, starting from 1.

The given sequence is 1, 2, 3, 4, 5, 6, 7.

We add 1 to the previous number to get the next one.

$7 + 1 = 8$

$8 + 1 = 9$

$9 + 1 = 10$

$\boxed{\text{The next three numbers are 8, 9, 10. The rule is to add 1 to the previous number.}}$

Step 3 — Odd numbers

This sequence lists odd numbers in order, starting from 1.

The given sequence is 1, 3, 5, 7, 9, 11.

We add 2 to the previous number to get the next one.

$11 + 2 = 13$

$13 + 2 = 15$

$15 + 2 = 17$

$\boxed{\text{The next three numbers are 13, 15, 17. The rule is to add 2 to the previous number.}}$

Step 4 — Even numbers

This sequence lists even numbers in order, starting from 2.

The given sequence is 2, 4, 6, 8, 10, 12, 14.

We add 2 to the previous number to get the next one.

$14 + 2 = 16$

$16 + 2 = 18$

$18 + 2 = 20$

$\boxed{\text{The next three numbers are 16, 18, 20. The rule is to add 2 to the previous number.}}$

Step 5 — Triangular numbers

This sequence is formed by adding counting numbers.

The given sequence is 1, 3, 6, 10, 15, 21, 28.

We add the next counting number to the previous term. After adding 7 to get 28, we add 8, then 9, then 10.

$28 + 8 = 36$

$36 + 9 = 45$

$45 + 10 = 55$

$\boxed{\text{The next three numbers are 36, 45, 55. The rule is to add the next natural number in sequence.}}$

Step 6 — Squares

This sequence is formed by multiplying a counting number by itself.

The given sequence is 1, 4, 9, 16, 25, 36, 49. These are $1 \times 1$, $2 \times 2$, $3 \times 3$, $4 \times 4$, $5 \times 5$, $6 \times 6$, $7 \times 7$.

We need to find the next three squares. These will be $8 \times 8$, $9 \times 9$, $10 \times 10$.

$8 \times 8 = 64$

$9 \times 9 = 81$

$10 \times 10 = 100$

$\boxed{\text{The next three numbers are 64, 81, 100. The rule is to square the next natural number.}}$

Step 7 — Cubes

This sequence is formed by multiplying a counting number by itself three times.

The given sequence is 1, 8, 27, 64, 125, 216. These are $1 \times 1 \times 1$, $2 \times 2 \times 2$, $3 \times 3 \times 3$, $4 \times 4 \times 4$, $5 \times 5 \times 5$, $6 \times 6 \times 6$.

We need to find the next three cubes. These will be $7 \times 7 \times 7$, $8 \times 8 \times 8$, $9 \times 9 \times 9$.

$7 \times 7 \times 7 = 343$

$8 \times 8 \times 8 = 512$

$9 \times 9 \times 9 = 729$

$\boxed{\text{The next three numbers are 343, 512, 729. The rule is to cube the next natural number.}}$

Step 8 — Virahanka numbers

This sequence is also known as the Fibonacci sequence.

The given sequence is 1, 2, 3, 5, 8, 13, 21.

We add the two previous numbers to get the next one. For example, $1 + 2 = 3$, $2 + 3 = 5$, $3 + 5 = 8$, and so on.

$13 + 21 = 34$

$21 + 34 = 55$

$34 + 55 = 89$

$\boxed{\text{The next three numbers are 34, 55, 89. The rule is to add the last two numbers.}}$

Step 9 — Powers of 2

This sequence is formed by multiplying the previous number by 2.

The given sequence is 1, 2, 4, 8, 16, 32, 64.

We multiply the last number by 2 to get the next one.

$64 \times 2 = 128$

$128 \times 2 = 256$

$256 \times 2 = 512$

$\boxed{\text{The next three numbers are 128, 256, 512. The rule is to multiply the last number by 2.}}$

Step 10 — Powers of 3

This sequence is formed by multiplying the previous number by 3.

The given sequence is 1, 3, 9, 27, 81, 243, 729.

We multiply the last number by 3 to get the next one.

$729 \times 3 = 2187$

$2187 \times 3 = 6561$

$6561 \times 3 = 19683$

$\boxed{\text{The next three numbers are 2187, 6561, 19683. The rule is to multiply the last number by 3.}}$

Answer

All 1's: 1, 1, 1, 1, 1, 1, 1. The next three numbers are 1, 1, 1. The rule is that every number is 1. Counting numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. The next three numbers are 8, 9, 10. The rule is to add 1 to the previous number. Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17. The next three numbers are 13, 15, 17. The rule is to add 2 to the previous number. Even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. The next three numbers are 16, 18, 20. The rule is to add 2 to the previous number. Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55. The next three numbers are 36, 45, 55. The rule is to add the next natural number in sequence. Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The next three numbers are 64, 81, 100. The rule is to square the next natural number. Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729. The next three numbers are 343, 512, 729. The rule is to cube the next natural number. Virahanka numbers: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The next three numbers are 34, 55, 89. The rule is to add the last two numbers. Powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512. The next three numbers are 128, 256, 512. The rule is to multiply the last number by 2. Powers of 3: 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683. The next three numbers are 2187, 6561, 19683. The rule is to multiply the last number by 3.