Diamond Pattern: Number of Diamonds in Each Diagram

The diagram below shows a sequence of diamond patterns.

1. The information from the sequence of diamonds is tabulated below. Complete the table.

   Pattern \({n}\)	Formula	Number of diamonds
   \({1}\)	\({1}\)	\({1}\)
   \({2}\)	\({4 + 1}\)	\({5}\)
   \({3}\)	\({4 + 4 + 1}\)	\({9}\)
   \({4}\)
   \({5}\)

2. Write down a formula for the \({n}^{\mathrm{th}}\) pattern.
3. Hence or otherwise, find the number of diamonds in the \({25}^{\mathrm{th}}\) pattern.
4. In pattern \({k}\), the number of diamonds is \({509}\). Find \({k}\).