Secondary 4 sequence questions become easier when students separate the pattern from the formula that describes it. This Mathematics tuition guide for Punggol families explains common differences, nth-term rules, checking formulas and more complex number patterns through original worked examples.

A student may be able to continue 5, 8, 11, 14 but become unsure when asked for the nth term. Another may guess a formula that works for the first two terms but fails later. The useful skill is not pattern spotting alone; it is building and checking a rule.

At eduKatePunggol, Secondary 4 Mathematics is taught in focused groups of up to three students in 1.5-hour lessons near Punggol MRT. This guide supports the wider Secondary 4 Mathematics year plan. Match practice to the student’s subject level and school syllabus.

Arithmetic sequences have a constant difference

Consider:

5, 8, 11, 14, …

The common difference is 3.

A linear nth-term rule therefore has coefficient 3 in front of n.

Worked example 1: find the nth term

For 5, 8, 11, 14, … start with 3n.

When n = 1, 3n gives 3, but the first term should be 5. Add 2.

nth term = 3n + 2.

Check n = 4:

3(4) + 2 = 14.

A rule should be checked against several terms, not accepted after one match.

Worked example 2: decreasing sequence

Find the nth term of:

20, 16, 12, 8, …

The common difference is −4, so begin with −4n.

At n = 1, −4n gives −4. To reach 20, add 24.

nth term = 24 − 4n.

Check n = 3: 24 − 12 = 12.

Use the nth term to test membership

Suppose the nth term is 4n + 1. Is 61 in the sequence?

Solve:

4n + 1 = 61
4n = 60
n = 15.

Because n = 15 is a positive whole-number term position, 61 is in the sequence.

If the solution had been n = 15.5, the number would not occur as a term in this sequence.

Worked example 3: find a particular term efficiently

If the nth term is 7n − 4, find the 30th term.

7(30) − 4 = 206.

There is no need to write out the first 30 terms once the general rule is known.

Second differences can reveal a quadratic pattern

Consider:

2, 6, 12, 20, 30, …

First differences are 4, 6, 8, 10. The second differences are constant at 2.

This suggests a quadratic nth-term rule.

Worked example 4: recognise a simple quadratic pattern

Notice that:

• 1 × 2 = 2;
• 2 × 3 = 6;
• 3 × 4 = 12;
• 4 × 5 = 20.

Therefore the nth term is:

n(n + 1) = n² + n.

Checking n = 5 gives 5 × 6 = 30.

The student does not need to force every quadratic pattern through one memorised algorithm if a clear structural pattern is visible.

A diagram pattern should be counted systematically

When a question shows a growing arrangement of tiles, match the diagram number n with the number of objects.

• What part grows with n?
• What part stays fixed?
• Does each new figure add a constant amount?
• Is there a rectangle, square or repeated strip hidden in the picture?

Turning the visual structure into algebra is more reliable than guessing from three numerical terms alone.

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How we diagnose sequence mistakes

Difference error: the student does not check whether the difference is constant.

Index-position error: n is confused with the term value.

Rule-check error: a guessed rule is not tested against later terms.

Membership error: a non-integer n is accepted as a valid term position.

Pattern-structure error: a diagram is counted visually without identifying the part that changes.

Why the three-student format helps

In a group of up to three students, one learner can explain the common difference, another can derive the nth term and another can test the rule. The tutor can see whether the formula is understood or merely copied.

What a 90-minute lesson could look like

An illustrative lesson could begin with ten minutes of quick pattern retrieval, twenty minutes on arithmetic nth terms, twenty minutes on testing membership, twenty minutes on more complex patterns and twenty minutes for independent diagram questions, error review and continuation work.

Repair, stabilisation and extension

Repair: use simple arithmetic sequences with positive common differences and check several terms.

Stabilisation: mix increasing and decreasing sequences, term-position questions and membership tests.

Extension: use quadratic or visual patterns and require students to explain why the rule matches the construction.

Try a short independent set

• Find the nth term of 7, 11, 15, 19, …
• Find the 25th term of 5n − 2.
• Is 83 in the sequence 6n − 1?

Answers: 4n + 3; 123; and yes, because 6n − 1 = 83 gives n = 14.

What progress should look like

• common differences are identified correctly;
• n is understood as a term position;
• rules are checked against multiple terms;
• membership requires an appropriate whole-number position;
• diagram patterns are decomposed structurally;
• students can explain why the rule works.

Punggol class details and consultation inputs

eduKatePunggol Secondary Mathematics tutorials run for 1.5 hours in groups of up to three students near Punggol MRT. Confirm current class availability, fees and meeting arrangements directly.

Bring the student’s subject level, examination year and recent sequence questions, especially any diagram-pattern questions where the student could continue the pattern but could not form the nth term.

Frequently asked questions

What does n mean?

n represents the position of a term: n = 1 for the first term, n = 2 for the second, and so on.

How do I know a number is not in a sequence?

Set the nth-term formula equal to that number and solve for n. If the resulting n is not a permitted term position, the number is not in the sequence.

Find the rule, then prove it works

Return to the Secondary 4 Mathematics year plan for the wider revision runway. For graph-based relationships, see the functions and graphs guide.

Observe the change, build the rule and test it beyond the first term. Families can WhatsApp eduKatePunggol with recent school work to discuss a suitable next step.