Figurate numbers can be represented by dots arranged in the shape of certain geometric figures. For example, the 1st four square numbers are shown below.

1. The 1st four triangular numbers are shown below.
   1. Determine the next three triangular numbers by drawing the corresponding triangles.
   2. There are only two numbers less than 100 that are both triangular and square. Find them.
   3. There is only one number between 100 and 10,000 that is both triangular and square. Find it. (Hint: Search the World Wide Web.)
   4. In one of our examples from class we actually computed the 100th triangular number. What is the 100th triangular number and how did we compute it?
2. Consider the following sequence of figurate numbers.

   1,6,15,28,45,66,...

   What geometric figures correspond to the numbers in this sequence? What is the 100th number in this sequence?
3. The nth term in the sequence of square numbers is n². To the ancient Greeks, the square root of a number in this sequence was the number of dots along one side of the square that represents the number. For non-square natural numbers, they used a clever technique to estimate the square roots. This technique is illustrated below. Use this technique to arrive at the approximation [Ö22] » 4⁶/₉.
4. Notice that for any natural number A, ÖA is the unique positive solution of the equation x² = A. So the Greek method for approximating square roots doubles as a method for solving equations of the form x² = A. For example, if we refer to the problem above, we see that the unique positive solution of x² = 22 is approximately 4 ⁶/₉.
   1. Now consider the sequence of rectangular numbers whose 1st three terms are shown below. Find the next four terms of this sequence.
   2. The nth term of this sequence is n²+n. Verify this by evaluating n²+n for n = 1,2,...,7 and comparing your results with those in part (a).
   3. We can now use a technique similar to that introduced in Problem 3 to approximate the unique positive solution of the equation x²+x = A, where A is some natural number. For example, the unique positive solution of x²+x = 9 is approximately 2 ³/₆, as illustrated below. Use this method to approximate the unique positive solution of x²+x = 22.
5. Consider the sequence whose nth term is n²+2n.
   1. Write out the 1st seven terms of this sequence.
   2. This is also a sequence of rectangular numbers. Draw the 1st three rectangles that correspond to these numbers.
   3. These rectangles can be used to approximate the unique positive solution of the equation x²+2x = A, where A is some natural number. Use this technique to approximate the unique positive solution of x²+2x = 13.