In the diagram below the pattern with 1 triangle needs 3 rods. The pattern with 2 triangles needs 5 rods.

Try to answer the following questions using the diagram above.

- Question
Complete the table for the number of rods needed for different numbers of triangles.

Number of triangles (T) 1 2 3 4 5 6 Number of rods (R) 3 5 The second row of the table should read 3, 5, 7, 9, 11, 13.

Number of triangles (T) 1 2 3 4 5 6 Number of rods (R) 3 5 7 9 11 13

- Question
Write down a formula for the number of rods, R, needed in a pattern with T triangles.

R is rising by 2 each time. This starts the formula with \(R = 2T\), but in order for this formula to work for any number of triangles, we need to add 1.

So the formula is:

\[R = 2T + 1\]

- Question
How many rods are needed for a pattern with 12 triangles?

\[R = 2 \times 12 + 1 = 25\,rods\]

- Question
How many triangles are there in a pattern using 49 rods?

\[49 = 2T + 1\]

\[2T = 48\]

\[T = 24\]

The number of triangles needed is 24.