The triangular numbers are the numbers 1, 3, 6, 10, 15, . . .; the square numbers are the numbers

1, 4, 9, 16, 25, . . . The pentagonal numbers 1, 5, 12, 22, 35, . . .. The geometrical language is
justified by the following diagrams:
a. What are the first five hexagonal numbers? What are the first five septagonal numbers?
What are the first five r-gonal numbers? Give a formula for the nth triangular number.
Give a formula for the nth square number. Give a formula for the nth pentagonal number.
In general, give a formula for the nth r-gonal number.
b. How many numbers can you find that are simultaneously triangular and square? How
many numbers can you find that are simultaneously square and pentagonal?
10. The tail of a giant kangaroo is attached by a giant rubber band to a stake in the ground. A

a)

- The first five hexagonal numbers can be found by using the formula: H(n) = n(2n-1)
Plugging in values for n, we get: H(1) = 1, H(2) = 6, H(3) = 15, H(4) = 28, H(5) = 45
- The first five septagonal numbers can be found by using the formula: S(n) = n(5n-3)/2
Plugging in values for n, we get: S(1) = 1, S(2) = 7, S(3) = 16, S(4) = 28, S(5) = 43
- The first five r-gonal numbers can be found by using the formula: P(n) = n((r-2)n-(r-4))/2
Plugging in values for r and n, we can obtain the respective formulas for different polygons:
- Triangular numbers: P(n) = n(n+1)/2
- Square numbers: P(n) = n^2
- Pentagonal numbers: P(n) = n(3n-1)/2
- Hexagonal numbers: P(n) = n(2n-1)
- Septagonal numbers: P(n) = n(5n-3)/2

b)
- To find numbers that are simultaneously triangular and square, we can look for numbers that satisfy both the triangular number formula and the square number formula. By comparing the formulas, we can set up the equation:
n(n+1)/2 = n^2
Simplifying the equation, we get:
n^2 + n = 2n^2
Rearranging, we have:
n^2 - n = 0
Factoring out n, we get:
n(n - 1) = 0
Therefore, there are 2 numbers that are simultaneously triangular and square: n = 0 and n = 1.

- To find numbers that are simultaneously square and pentagonal, we can set up a similar equation using the square number formula and pentagonal number formula:
n^2 = n(3n-1)/2
Simplifying the equation, we get:
2n^2 = 3n^2 - n
Rearranging, we have:
n^2 - n = 0
Factoring out n, we get:
n(n - 1) = 0
Therefore, there are 2 numbers that are simultaneously square and pentagonal: n = 0 and n = 1.

10. The given question about the giant kangaroo seems to be incomplete. Could provide the complete question?