P6 Math Pattern Questions: Master Figure 100 & The dn+c Formula

Why Pattern Questions Trip Up P6 Students

Of all the question types in the PSLE Math paper, number pattern questions (often called "Figure" questions) are the ones that most students attempt using the slow, painful method: drawing the figure and counting.

For Figure 10, that's fine. For Figure 100, drawing is impossible—and that's exactly why PSLE examiners love asking it.

This guide will give you the formula-based approach that eliminates drawing forever.

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Understanding the Pattern Question Structure

Most P6 pattern questions follow this format:

"Below is a sequence of figures made from squares. Figure 1 has 5 squares. Figure 2 has 8 squares. Figure 3 has 11 squares. How many squares are in Figure 100?"

To solve this without drawing:

1. ▸

   Find the Gap (Common Difference): How many squares are added each step?

   • ▸Figure 1 → Figure 2: $8 - 5 = 3$
   • ▸Figure 2 → Figure 3: $11 - 8 = 3$
   • ▸The gap is 3.
2. ▸

   Use the Formula: $T(n) = d \times n + c$

   • ▸$d$ = the gap (common difference) $= 3$
   • ▸$n$ = the figure number
   • ▸$c$ = the adjustment constant
3. ▸

   Find $c$: Substitute a known value.

   • ▸$T(1) = 5$, so: $3 \times 1 + c = 5 \rightarrow c = 2$
4. ▸

   Answer Figure 100:

   • ▸$T(100) = 3 \times 100 + 2 = 302$ squares

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The dn+c Formula: A Closer Look

Why It's Called dn+c

The formula is actually the same as the algebraic linear equation $y = mx + b$. At P6 level, MOE uses $dn+c$ notation to keep it pre-algebra friendly.

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The "Figure 0" Shortcut

Here's a faster way to find c that many top students use:

Work backwards from Figure 1 to find what Figure 0 would be.

Using our example:

• ▸Figure 1 has 5 squares, gap is 3
• ▸Figure 0 would have $5 - 3 = 2$ squares

That 2 is your $c$ value. Always.

This shortcut works because $c = T(0)$ in the formula, i.e., the "starting value" before the pattern begins.

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Common PSLE Pattern Question Types

Type 1: Linear Number Patterns

The simplest type—a constant gap between each term.

Example: $4, 7, 10, 13, \ldots$ What is the 50th term?

• ▸Gap $(d) = 3$
• ▸$c = 4 - 3 = 1$
• ▸$T(50) = 3 \times 50 + 1 = 151$

Type 2: Figure Patterns (Sticks or Squares)

Requires counting from a diagram, then applying $dn+c$.

Tip: Always verify by checking your formula against TWO known values (not just one).

Type 3: Two-Variable Patterns (Advanced)

Some P6 questions give two different properties of the figure (e.g., squares AND perimeter). Solve each property separately with its own $dn+c$ formula.

Type 4: "How Many Figures Have Less Than X?"

Work backwards: solve for $n$ when $T(n) < X$.

Example: Using $T(n) = 3n + 2$, find the largest figure with fewer than 50 squares.

• ▸$3n + 2 < 50$
• ▸$3n < 48$
• ▸$n < 16$
• ▸Answer: Figure 15 (the largest figure with $\leq 47$ squares)

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Step-by-Step Problem Solving Framework

For every pattern question, use this 4-step framework:

Step 1 — Find the Gap Subtract consecutive terms. If the gap is constant, you have a linear pattern.

Step 2 — Find $c$ (using Figure 0 shortcut) Subtract one gap from the first term. This is your $c$.

Step 3 — Write the Formula $T(n) = d \times n + c$

Step 4 — Verify Plug in $n=2$ and $n=3$. Do you get the correct terms? If yes, proceed.

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Practice Questions

Practice Questions

Try these yourself using the $dn+c$ method:

Q1: A pattern produces $3, 8, 13, 18, \ldots$ What is the 60th term?

• ▸Step 1: Find the common difference: $d = 8 - 3 = 5$
• ▸Step 2: Find $c$ using $T(1) = 3$: $5(1) + c = 3 \rightarrow c = -2$
• ▸Step 3: Formula: $T(n) = 5n - 2$
• ▸Step 4: 60th term: $T(60) = 5(60) - 2 = 298$ (Answer: 298)

Q2: A figure pattern has 7 sticks in Figure 1 and 12 sticks in Figure 2. How many sticks in Figure 25?

• ▸Step 1: Find the common difference: $d = 12 - 7 = 5$
• ▸Step 2: Find $c$ using $T(1) = 7$: $5(1) + c = 7 \rightarrow c = 2$
• ▸Step 3: Formula: $T(n) = 5n + 2$
• ▸Step 4: Figure 25: $T(25) = 5(25) + 2 = 127$ (Answer: 127)

Q3: Using the pattern $T(n) = 4n + 3$, which is the first figure with more than 100 items?

• ▸Step 1: Solve $4n + 3 > 100 \rightarrow 4n > 97 \rightarrow n > 24.25$
• ▸Step 2: The smallest integer $n$ is 25
• ▸Step 3: Verify: $T(25) = 4(25) + 3 = 103 > 100$, while $T(24) = 4(24) + 3 = 99 \leq 100$ (Answer: Figure 25)

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Why This Formula Always Works

The $dn+c$ formula works because PSLE Math only tests linear patterns at the P6 level. Non-linear patterns (quadratic, geometric) are not part of the Primary syllabus. This means:

• ▸The gap between consecutive terms is always constant
• ▸The formula always fits
• ▸There is no exception at the PSLE level

Once a student understands this, pattern questions become the easiest 2-mark questions on the paper.

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Frequently Asked Questions

What is the formula for Figure 100 in math patterns?

To find Figure 100, use $T(n) = d \times n + c$. First find the common difference ($d$) by subtracting consecutive terms. Then find $c$ by working backwards to Figure 0. Finally, substitute $n=100$.

How do you find the common difference in a P6 pattern?

Subtract any term from the next term. For example, if Figure 2 has 8 items and Figure 1 has 5, the common difference is $8 - 5 = 3$.

What is the $dn+c$ formula for PSLE Math?

The $dn+c$ formula calculates the number of items in the $n$-th figure of a linear pattern. '$d$' is the common difference, '$n$' is the figure number, and '$c$' is the constant (what Figure 0 would contain).

Can all PSLE pattern questions be solved with $dn+c$?

Yes. All pattern questions at the P6/PSLE level involve linear sequences. The dn+c formula solves them all without drawing.