PSHS NCE Patterns, Sequences, and Number Relationships Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Patterns, Sequences, and Number Relationships

Spot how numbers change, find the rule, and predict what comes next without checking every possible answer.

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Patterns, Sequences, and Number Relationships

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Ask: what changes each time?

Sequence = numbers arranged in an order that follows a rule.

Same difference:
5, 9, 13, 17
Add 4 each time.

Same factor:
3, 6, 12, 24
Multiply by 2 each time.

Changing difference:
2, 5, 9, 14
Add 3, then 4, then 5.

Alternating rule:
4, 7, 14, 17, 34
+3, ×2, +3, ×2.

FAST METHOD

Check the gaps first

1. Subtract neighboring terms. Are the differences the same?
2. If not, check whether the differences themselves follow a pattern.
3. Check simple multiplication or division.
4. If neither works, test an alternating rule.
5. Use the simplest rule that explains every term.

WORKED EXAMPLES

Six common pattern types

1. Same difference

12, 17, 22, 27, ?

Add 5 each time.

32

2. Same factor

2, 6, 18, 54, ?

Multiply by 3 each time.

162

3. Growing differences

4, 7, 11, 16, 22, ?

Differences are +3, +4, +5, +6. Next is +7.

29

4. Alternating operations

5, 10, 12, 24, 26, ?

×2, +2, ×2, +2, so next is ×2.

52

5. Square numbers

1, 4, 9, 16, 25, ?

These are 1², 2², 3², 4², 5².

36

6. Missing middle term

8, 13, ?, 23, 28

Add 5 each time.

18

TRY WITH ME

Use the gaps

Find the next number:

3, 7, 12, 18, 25, ?

1. Find the differences: +4, +5, +6, +7.
2. The difference increases by 1 each time.
3. Next difference is +8.

33

COMMON TRAPS

Watch out for these

Guessing from only the first two terms.
The rule must fit the whole sequence.

Missing an alternating pattern.
Try checking odd-position and even-position steps separately.

Forcing a complicated rule.
Entrance tests usually reward the simplest consistent pattern.

Ignoring changing differences.
If the first differences are not equal, inspect how they change.

REGULAR PRACTICE

Build the skill

QUICK FIRE

Can you spot the rule fast?

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency.

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Can't see the pattern yet?

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Arithmetic Sequences: nth Term, Common Difference and UPCAT Practice

TEACHER ABI UPCAT MATHEMATICS

Arithmetic Sequences

Recognize constant differences, move efficiently between terms, and translate sequence information into equations.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Arithmetic sequences change by a constant difference

An arithmetic sequence is an ordered list of numbers in which the same amount is added or subtracted each time. That fixed amount is the common difference, written as d.

For example, in 8, 13, 18, 23, ..., every term increases by 5, so d=5.

The nth-term formula

aₙ = a₁ + (n−1)d

aₙ is the term you want, a₁ is the first term, n is its position, and d is the common difference. The expression n−1 counts how many jumps occur after the first term.

See the formula in action

Find the 4th term of 8, 13, 18, 23, ...

a₄ = 8 + (4−1)(5) = 8 + 15 = 23.

This matches the visible jumps: 8 → 13 → 18 → 23. Reaching the 4th term requires three jumps of 5.

Common difference

Subtract consecutive terms in the same order: later term minus earlier term.

nth term

The first term needs zero jumps; the nth term needs n−1 jumps.

DO IT FAST

Count jumps, not term numbers

From term aᵣ to term aₙ, there are n−r jumps. Use aₙ=aᵣ+(n−r)d. This avoids restarting from the first term.

Why it works

Each jump adds d exactly once. The number of additions is the difference between the two term positions.

WORKED EXAMPLES

Five forms you should recognize

1. Find d

7,11,15,... has d=4.

2. Find an nth term

a₁=8,d=5: a₁₅=8+14(5)=78.

3. Find a missing term

5,__,13 has equal gaps, so the middle term is 9.

4. Use two known terms

a₄=14 and a₉=34: five jumps total 20, so d=4.

5. Model a pattern

A theater row with 18 seats, then 22, then 26 follows aₙ=18+4(n−1).

COMMON TRAPS

Check before you commit

• Using n instead of n−1: the first term has made zero jumps.
• Changing differences: verify at least two consecutive gaps.
• Wrong subtraction order: later minus earlier keeps the sign meaningful.
• Confusing term and value: “which term is 50?” asks for n.
• Assuming arithmetic: constant ratios indicate a geometric sequence instead.
• Ignoring position gaps: a₉−a₄ covers five, not nine, jumps.

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Arithmetic Sequences FAQ

How do I know a sequence is arithmetic?

Its consecutive differences are constant.

Why is the formula n−1?

The first term is already present, so reaching term n requires n−1 additions.

Can d be negative?

Yes. A decreasing arithmetic sequence has a negative common difference.

How do I find which term equals a value?

Set the nth-term expression equal to that value and solve for n.

RELATED COMPETENCIES

Continue your mathematics review.

Pattern RecognitionLinear FunctionsGeometric SequencesAlgebraic EquationsSeries

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