Number Patterns and Sequences

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A sequence is an ordered list of numbers that follow a specific pattern or rule. Being able to identify the pattern — and continue it — is a fundamental skill tested in the NCEE quantitative reasoning section.

Common Sequence Types:

1. Arithmetic Sequence (Linear Pattern): Each term increases or decreases by a constant amount (the common difference $d$).

Pattern: $a, a+d, a+2d, a+3d, …$

Example: $5, 8, 11, 14, 17, …$

• First term ($a$) = 5
• Common difference ($d$) = 3
• Next term: $17 + 3 = 20$

Formula for the $n^{th}$ term: $T_n = a + (n-1)d$

2. Geometric Sequence (Exponential Pattern): Each term is multiplied or divided by a constant (the common ratio $r$).

Pattern: $a, ar, ar^2, ar^3, …$

Example: $3, 9, 27, 81, 243, …$

• First term ($a$) = 3
• Common ratio ($r$) = 3
• Next term: $243 \times 3 = 729$

Formula for the $n^{th}$ term: $T_n = ar^{n-1}$

3. Square Numbers (Perfect Squares): $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …$

Pattern: $T_n = n^2$ — each number is the square of its position

4. Cube Numbers: $1, 8, 27, 64, 125, …$

Pattern: $T_n = n^3$

5. Triangular Numbers: $1, 3, 6, 10, 15, 21, 28, 36, …$

Pattern: Each term = sum of first $n$ natural numbers: $T_n = n(n+1)/2$

6. Fibonacci-like Sequences: Each term is the sum of the two terms before it. $1, 1, 2, 3, 5, 8, 13, 21, 34, …$ (Each number = sum of the two before it)

⚡ Exam Tip (NCEE): Always check whether the pattern is ADDITION (constant difference → arithmetic) or MULTIPLICATION (constant ratio → geometric). For complex sequences, check: (1) is it arithmetic? (2) is it geometric? (3) are there squares/cubes? (4) is it a combination of patterns?

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of pattern recognition.

Identifying the Pattern — Systematic Approach:

1. Look at the differences between consecutive terms

   • If the difference is constant → Arithmetic sequence
   • Example: $7, 12, 17, 22$ → difference of 5 each time

2. Look at the ratio between consecutive terms

   • If the ratio is constant → Geometric sequence
   • Example: $3, 6, 12, 24$ → ratio of 2 each time

3. Look at whether terms are perfect squares or cubes

   • $1, 4, 9, 16, 25$ → These are $1^2, 2^2, 3^2, 4^2, 5^2$

4. Look for combinations

   • Some sequences alternate between two patterns or add a second layer

Worked Examples:

Example 1 — Arithmetic: “Find the 10th term of the sequence: $2, 9, 16, 23, …$”

$d = 9 - 2 = 7$ $T_{10} = 2 + (10-1) \times 7 = 2 + 63 = 65$

Example 2 — Arithmetic: “Find the 15th term of: $5, 11, 17, 23, …$”

$d = 6$ $T_{15} = 5 + (15-1) \times 6 = 5 + 84 = 89$

Example 3 — Geometric: “Find the 6th term of: $4, 12, 36, 108, …$”

$r = 12/4 = 3$ $T_6 = 4 \times 3^{6-1} = 4 \times 3^5 = 4 \times 243 = 972$

Example 4 — Mixed Pattern: “3, 5, 9, 17, 33, …” Differences: $2, 4, 8, 16$ → Each difference is doubling The next difference = $16 \times 2 = 32$ Next term: $33 + 32 = 65$

Finding Missing Terms:

Example: Find the missing number in: $4, __, 16, 25, 36$

These are square numbers: $2^2, __, 4^2, 5^2, 6^2$ Missing = $3^2 = 9$

Sum of Arithmetic Sequence:

Sum of first $n$ terms: $S_n = n/2 \times (first term + last term)$ OR: $S_n = n/2 \times [2a + (n-1)d]$

Example: Sum of $2 + 9 + 16 + 23 + 30$ ($n=5$) $S_5 = 5/2 \times (2 + 30) = 5/2 \times 32 = 80$

⚡ Common NCEE Error: When asked to find the $n^{th}$ term, students often use the wrong formula. Remember: the formula $T_n = a + (n-1)d$ is for arithmetic sequences. For geometric sequences, it is $T_n = ar^{n-1}$. Always identify the sequence type first.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Arithmetic Mean:

The arithmetic mean of two numbers $a$ and $b$ is $(a+b)/2$. This is the number that, when added to a sequence, creates an arithmetic sequence.

If $a, AM, b$ are in arithmetic progression: $AM = (a+b)/2$

Geometric Mean:

The geometric mean of two numbers $a$ and $b$ is $\sqrt{ab}$. If $a, GM, b$ are in geometric progression: $GM = \sqrt{ab}$

Harmonic Sequence:

A sequence where the reciprocals form an arithmetic sequence. Example: $1, 1/2, 1/3, 1/4, 1/5, …$

The harmonic mean of $a$ and $b$ is: $HM = 2ab/(a+b)$

Relationship between AM, GM, and HM: For any two positive numbers: $AM \geq GM \geq HM$ Equality holds only when $a = b$.

Applications in NCEE-Style Problems:

Pattern Recognition in Real Contexts:

1. Calendar patterns:

   • Days of the week repeat every 7 days
   • If today is Monday, the 25th day is: $25 \mod 7 = 4$ → Thursday

2. Clock patterns:

   • Hour hand moves 30° per hour
   • Minute hand moves 6° per minute

3. Seating arrangements:

   • If people sit in a circle, arrangements relative to each other matter

Recurring Decimals as Sequences:

$0.333… = 3/10 + 3/100 + 3/1000 + … = 1/3$ $0.999… = 0.9 + 0.09 + 0.009 + … = 1$ (geometric series with $r = 0.1$)

Pascal’s Triangle — Pattern Recognition:

Row 0:         1
Row 1:        1 1
Row 2:       1 2 1
Row 3:      1 3 3 1
Row 4:     1 4 6 4 1

• Each row starts and ends with 1
• Each interior number is the sum of the two numbers above it
• Row $n$ gives binomial coefficients for $(a+b)^n$
• The sum of row $n$ = $2^n$

Prime Numbers — Pattern Recognition:

Prime numbers: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …$

• 2 is the only even prime
• All other primes are odd
• Every composite number can be written as a product of primes (Fundamental Theorem of Arithmetic)

Prime Number Tests:

• A number divisible by 2: last digit is even
• Divisible by 3: sum of digits divisible by 3
• Divisible by 5: last digit is 0 or 5

⚡ Extended Tip — Generalising Patterns: When a sequence doesn’t follow a simple arithmetic or geometric rule, look for:

1. Second differences: If first differences aren’t constant, check if second differences (differences of differences) are constant → quadratic pattern
2. Multiplying by increasing factors: $2, 6, 18, 54$ is $2 \times 3^0, 2 \times 3^1, 2 \times 3^2, 2 \times 3^3$ — a geometric sequence with first term 2
3. Alternating operations: $1, 3, 4, 12, 13, 39$ → $+2, \times3, +2, \times3, …$

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