Patterns in Mathematics | FIO

Question 4

Rewrite each sequence of Table 1 in your notebook, along with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence.

Solution

Understand the Question

A number sequence is an ordered list of numbers that follows a specific rule or pattern.
To find the next three numbers in each sequence:
1. Identify the relationship between consecutive terms (e.g., adding a constant value, multiplying by a factor, adding consecutive integers, or summing the previous two terms).
2. Apply that rule repeatedly to generate the subsequent terms.
3. State the rule clearly in words.

Step 1 · All 1's

Given sequence: $1, 1, 1, 1, \dots$

\begin{aligned} 1 \\ 1 \\ 1 \end{aligned}

Next three numbers: $1, 1, 1$
Rule: Every number in the sequence is $1$ (a constant sequence).

Step 2 · Counting Numbers

Given sequence: $1, 2, 3, 4, 5, 6, 7, \dots$

\begin{aligned} 7 + 1 &= 8 \\ 8 + 1 &= 9 \\ 9 + 1 &= 10 \end{aligned}

Next three numbers: $8, 9, 10$
Rule: Add $1$ to the previous number.

Step 3 · Odd Numbers

Given sequence: $1, 3, 5, 7, 9, 11, \dots$

\begin{aligned} 11 + 2 &= 13 \\ 13 + 2 &= 15 \\ 15 + 2 &= 17 \end{aligned}

Next three numbers: $13, 15, 17$
Rule: Start at $1$ and add $2$ to the previous number.

Step 4 · Even Numbers

Given sequence: $2, 4, 6, 8, 10, 12, 14, \dots$

\begin{aligned} 14 + 2 &= 16 \\ 16 + 2 &= 18 \\ 18 + 2 &= 20 \end{aligned}

Next three numbers: $16, 18, 20$
Rule: Start at $2$ and add $2$ to the previous number.

Step 5 · Triangular Numbers

Given sequence: $1, 3, 6, 10, 15, 21, 28, \dots$

The differences between consecutive terms increase by $1$ each time ( $+2, +3, +4, +5, +6, +7, \dots$ ).

\begin{aligned} 28 + 8 &= 36 \\ 36 + 9 &= 45 \\ 45 + 10 &= 55 \end{aligned}

Next three numbers: $36, 45, 55$
Rule: Add the next natural number to the previous term.

Step 6 · Squares

Given sequence: $1, 4, 9, 16, 25, 36, 49, \dots$

These terms are $1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2$ .

\begin{aligned} 8 \times 8 &= 64 \\ 9 \times 9 &= 81 \\ 10 \times 10 &= 100 \end{aligned}

Next three numbers: $64, 81, 100$
Rule: Multiply the position number ( $n$ ) by itself ( $n^2$ ).

Step 7 · Cubes

Given sequence: $1, 8, 27, 64, 125, 216, \dots$

These terms are $1^3, 2^3, 3^3, 4^3, 5^3, 6^3$ .

\begin{aligned} 7 \times 7 \times 7 &= 343 \\ 8 \times 8 \times 8 &= 512 \\ 9 \times 9 \times 9 &= 729 \end{aligned}

Next three numbers: $343, 512, 729$
Rule: Multiply the position number ( $n$ ) by itself three times ( $n^3$ ).

Step 8 · Virāhāṅka (Fibonacci) Numbers

Given sequence: $1, 2, 3, 5, 8, 13, 21, \dots$

Each term is obtained by adding the preceding two terms.

\begin{aligned} 13 + 21 &= 34 \\ 21 + 34 &= 55 \\ 34 + 55 &= 89 \end{aligned}

Next three numbers: $34, 55, 89$
Rule: Add the two previous numbers together.

Step 9 · Powers of 2

Given sequence: $1, 2, 4, 8, 16, 32, 64, \dots$

\begin{aligned} 64 \times 2 &= 128 \\ 128 \times 2 &= 256 \\ 256 \times 2 &= 512 \end{aligned}

Next three numbers: $128, 256, 512$
Rule: Multiply the previous number by $2$ .

Step 10 · Powers of 3

Given sequence: $1, 3, 9, 27, 81, 243, 729, \dots$

\begin{aligned} 729 \times 3 &= 2187 \\ 2187 \times 3 &= 6561 \\ 6561 \times 3 &= 19683 \end{aligned}

Next three numbers: $2187, 6561, 19683$
Rule: Multiply the previous number by $3$ .

Answer

All 1's: $1, 1, 1$ (Every number is $1$ )

Counting numbers: $8, 9, 10$ (Add $1$ to the previous number)
Odd numbers: $13, 15, 17$ (Add $2$ to the previous number)
Even numbers: $16, 18, 20$ (Add $2$ to the previous number)
Triangular numbers: $36, 45, 55$ (Add consecutive natural numbers $+8, +9, +10$ )
Squares: $64, 81, 100$ (Square the consecutive natural numbers $8^2, 9^2, 10^2$ )
Cubes: $343, 512, 729$ (Cube the consecutive natural numbers $7^3, 8^3, 9^3$ )
Virāhāṅka numbers: $34, 55, 89$ (Add the last two numbers)
Powers of 2: $128, 256, 512$ (Multiply the previous number by $2$ )
Powers of 3: $2187, 6561, 19683$ (Multiply the previous number by $3$ )

Common Mistakes

Confusing Addition with Multiplication: Mistaking geometric progressions like powers of $2$ ( $1, 2, 4, 8, \dots$ ) for arithmetic progressions like even numbers ( $2, 4, 6, 8, \dots$ ).
Triangular Numbers Step Difference: Forgetting that the amount added increases by $1$ for each subsequent term (e.g., $+2, +3, +4, \dots$ ) rather than adding a fixed constant.
Virāhāṅka / Fibonacci Terms: Adding a fixed number instead of adding the two immediately preceding terms.

Question 4

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