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

Series of rod triangles in groups, increasing from 1 to 5

Try to answer the following questions using the diagram above.


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

Number of triangles (T)123456
Number of rods (R)35

The second row of the table should read 3, 5, 7, 9, 11, 13.

Number of triangles (T)123456
Number of rods (R)35791113


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:

\[R = 2T + 1\]


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

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


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.