Problem of the Week #18: Monday May 8th, 2023 Sorry for the last update today. I was sick over the weekend and slept most of today away. As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems. Please register for an account if you would like to join the discussion below or share your own problems.
A triangular number is a natural number that can be expressed as the sum of more than one consecutive natural numbers starting from 1. For example, 10 is a triangular number because 10 = 1+2+3+4.
A trapezoidal number is a natural number that can be expressed as the sum of more than one consecutive natural numbers not including 1. For example, 9 is a trapezoidal number because 9 = 4+5 and 9 = 2+3+4. On the other hand, 10 is not a trapezoidal number because the only way to express 10 as the sum of consecutive natural numbers is to include 1 (as shown above).
a). Find a triangular number that is also a trapezoidal number.
b). Find a natural number that is neither a triangular number nor a trapezoidal number.
c). Find the first five natural numbers that are neither a triangle number nor a trapezoid number.
d). What do you notice about your answer in part c)? Does this pattern continue? Why or why not?
e). Notice that 9 can satisfy the definition of a trapezoidal number in two ways, and 7 can do so in only one way. In how many ways can 30 satisfy this definition?
f). In how many ways can a given natural number n satisfy this definition?
g). Find an expression for all triangular numbers that are trapezoidal numbers.
h). Share your own problem inspired by this one.
i). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.