A) To complete the table, we need to observe the patterns for the number of shaded, unshaded, and total triangles.

Mathematics

ScanSolve AI Answer

Answers

2 parts
b(i)

10\text{10}

b(ii)

4n - 2

Step-by-step solution

5 steps

Here's the solution for question 1:

Part a· 3 steps

To complete the table, we need to observe the patterns for the number of shaded, unshaded, and total triangles.

  1. Analyze the pattern for "Number of shaded triangles".

    Pattern 1: 2
    Pattern 2: 4
    Pattern 3: 6
    This is an arithmetic progression where the number of shaded triangles is 2×Pattern number2 \times \text{Pattern number}.
    For Pattern 4: 2×4=82 \times 4 = 8.

  2. Analyze the pattern for "Number of unshaded triangles".

    Pattern 1: 0
    Pattern 2: 2
    Pattern 3: 4
    Pattern 4: 6
    This is an arithmetic progression where the number of unshaded triangles is 2×(Pattern number−1)2 \times (\text{Pattern number} - 1).

  3. Analyze the pattern for "Total number of triangles".

    Pattern 1: 2
    Pattern 2: 6
    Pattern 3: 10
    This is an arithmetic progression with a common difference of 4.
    For Pattern 4: 10+4=1410 + 4 = 14.
    Alternatively, Total number of triangles = Number of shaded triangles + Number of unshaded triangles.
    For Pattern 4: 8+6=148 + 6 = 14.

    The completed table is:

    Pattern number1234Number of shaded triangles2468Number of unshaded triangles0246Total number of triangles261014\begin{array}{|l|c|c|c|c|} \hline \text{Pattern number} & 1 & 2 & 3 & 4 \\ \hline \text{Number of shaded triangles} & 2 & 4 & 6 & \textbf{8} \\ \hline \text{Number of unshaded triangles} & 0 & 2 & 4 & 6 \\ \hline \text{Total number of triangles} & 2 & 6 & 10 & \textbf{14} \\ \hline \end{array}

Part b(i)

10\text{10}

Find the number of unshaded triangles in pattern 6.

From the pattern observed in Step 2, the number of unshaded triangles is 2×(Pattern number−1)2 \times (\text{Pattern number} - 1).
For Pattern 6: 2×(6−1)=2×5=102 \times (6 - 1) = 2 \times 5 = 10.

The number of unshaded triangles in pattern 6 is 10\boxed{\text{10}}.

Part b(ii)4n - 2

Find the nth term, for the total number of triangles in pattern n.

From the table, the sequence for the total number of triangles is 2, 6, 10, 14, ...
This is an arithmetic progression with the first term a=2a = 2 and common difference d=6−2=4d = 6 - 2 = 4.
The formula for the nth term of an arithmetic progression is Tn=a+(n−1)dT_n = a + (n-1)d.

  1. Substitute the values of aa and dd into the formula.

    Tn=2+(n−1)4T_n = 2 + (n-1)4
  2. Simplify the expression.

    Tn=2+4n−4T_n = 2 + 4n - 4 Tn=4n−2T_n = 4n - 2

    The nth term for the total number of triangles in pattern n is 4n−2\boxed{4n - 2}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question