Triangular Number Problems
1. Prove that for any consecutive triangular numbers T(n-1) and T(n),
2. Find an closed form expression for the sum of the trianglular numbers, T(1) + T(2) + T(3) + . . . + T(n).
3. Note the sequence of triangle numbers begins 1, 3, 6, 10, 15, . . .
Write the beginning numbers for the sequence of SUMS of trianglular numbers.
4. Handshake problem.
An at meeting on 18 members of a club, each member shakes hands with every other member exactly once. What is the total number of handshakes? How many handshakes if there are 25 members in attendance?
5. When distinct 9 points are placed on a circle, how many segments are determined? (do not consider the intersections of the segments within the circle to determin new points and segments).
What if there are n points?
6. If we have n objects in a set selected 2 at a time, what is the number of combinations of n objects taken 2 at a time?
7. Examine the Pascal Triangle for a line of triangular numbers. Explain.
Note: Each number in the Pascal triangle is the sum of the two numbers immediately above it. Numbers at the end of a row are seen to have a 0 and 1 immediately above them.
8. Find a line in the Pascal Triangle showing the sequence of sums of triangular numbers. Explain.
9. Find an expression for the first 2n integers.
10. Find an expression for the first n even integers: 2 + 4 + 6 + 8 + . . . + 2n
11. Find an expression for the first n ODD integers: 1 + 3 + 5 + 7 + . . . + (2n-1).
Find several ways to show the sum of the first n odd integers.
For example:
For example:
Show the sum of first n odd integers is
T(2n) -2(T(n)
12. See Count the Triangles -- II
13. Find a reference for the "Rule of 78" -- a method that was used in lending to calculate yearly interst.
