parmenides51 wrote: Let $y$ be a positive integer, written in a $9$-digit base system of integers only. Show that $y$ is a triangular number, that is, for any positive integer $n$, the number $y$ is the sum of the n natural numbers from $1$ to $n$. Sorry, I don't seem to understand this problem. What does "written in a $9$-digit base system" mean? Is $y$ a positive integer written in base $9$? Is $y$ a $9$-digit number? Also, I think more information is needed in order to solve this problem.

$y$ is a positive integer written in base $9$ system, if that makes better sense, I shall correct it

parmenides51 wrote: $y$ is a positive integer written in base $9$ system, if that makes better sense, I shall correct it You mean $y$ consists of only digits from the set $\{0,1,2,3,\ldots,8\}$? Then, I think this is false. For example, consider $y=4$.

NikoIsLife wrote: parmenides51 wrote: $y$ is a positive integer written in base $9$ system, if that makes better sense, I shall correct it You mean $y$ consists of only digits from the set $\{0,1,2,3,\ldots,8\}$? Then, I think this is false. For example, consider $y=4$. parmenides51 wrote: Let $y$ be a positive integer, written in a 9-digit base system of integers only. Show that $y$ is a triangular number, that is, for any positive integer $n$, the number $y$ is the sum of the n natural numbers from $1$ to $n$. parmenides51 edited.

sonone wrote: NikoIsLife wrote: parmenides51 wrote: $y$ is a positive integer written in base $9$ system, if that makes better sense, I shall correct it You mean $y$ consists of only digits from the set $\{0,1,2,3,\ldots,8\}$? Then, I think this is false. For example, consider $y=4$. parmenides51 wrote: Let $y$ be a positive integer, written in a 9-digit base system of integers only. Show that $y$ is a triangular number, that is, for any positive integer $n$, the number $y$ is the sum of the n natural numbers from $1$ to $n$. parmenides51 edited. Can you please explain to me what the problem is trying to say? Sorry, I really have difficulty trying to understand this problem, but it seems like you know.

Suppose that y is a positive integer written only with digit 1, in base 9 system. Prove that y is a triangular number. Thanks for anybody helping to correct it, sorry for the mistakes, I ain't gonna try and translate from Finnish in the future using Google, I am gonna correct it when I use my pc

Write $$y = 1 + \cdots +9^n = (9^{n+1}-1)/8 = \frac{\left(\frac{3^{n+1}-1}{2}\right)\left(\frac{3^{n+1}-1}{2} + 1\right)}{2}$$$\square$ Remarks: A positive integer is triangular iff it can be written in the form $\frac{k(k+1)}{2}$ for natural $k$. Note that $\frac{3^{n+1}-1}{2} \in \mathbb{N}$, so our proof is watertight.