Problem 7. Find the sum of the first n triangular numbers (Hint: Show first that each triangular number can be n²+n represented as Tn 2 = = 1/10 2 n² + n). (

Problem 4. Let n be an integer. Show that if 2 + 2/1+ 12n2 is an integer, then it is the square of an integer.

1. Starting with any rectangle, we can create a new, larger rectangle by attaching a square to the longer side. For example, if we start with a 2 x 5 rectangle, we would glue on a 5 x 5 square, forming a 5 x 7 rectangle: The next rectangle would be formed by attaching a 7 x 7 square to the top or bottom of the 5 x 7 rectangle. (a) Create a sequence of rectangles using this rule starting with a 1 x 2 rectangle. Then write out the sequence of perimeters for the rectangles (the first term of the sequence would be 6, since the perimeter of a 1 x 2 rectangle is 6, the next term would be 10). (b) Repeat the above part this time starting with a 1 x 3 rectangle. (c) Find recursive definitions for each of the sequences of perimeters you found in parts (a) and (b). Don't forget to give the initial conditions as well. (d) Are the sequences arithmetic? Geometric? If not, are they close to being either of these (i.e., are the differences or ratios almost constant)? Write down the sequences of…