# Proving that the finite sum of the each reciprocal of any sequence of integers with common difference is not an integer.

Question : Could you show me how to prove that $\sum_{j=1}^{n}\frac{1}{a+jd}$ is not an integer for any integers $a\gt1, d\gt0$.

A week ago, I found the following question in a book:

Prove that $S_n=\sum_{k=2}^n \frac1k$ is not an integer for any $n$.

Proof : There exists only one integer $k$ such that $2^k\le n\lt 2^{k+1}$ for any $n$. Note that there exists an integer $t$ such that $\frac{s}{2^k}=t$ where $s$ is an integer which satisfies $s\le n$ only if $s=2^k$. Letting $T$ be the product of all odd numbers which are less than or equal to $n$, then we get $$2^{k-1}TS_n=\frac{2^{k-1}T}{2}+\frac{2^{k-1}T}{3}+\cdots+\frac{2^{k-1}T}{n}.$$ Hence, we know that $\frac{2^{k-1}T}{2^k}$ is not an integer and that the others of the right side are all integers. Hence, we know that $2^{k-1}TS_n$ is not an integer. Now, the proof is completed.

By the way, this book says, "It is known that $\sum_{j=1}^{n}\frac{1}{a+jd}$ is not an integer for any integers $a\gt1, d\gt0.$" without any additional information. I've tried to prove this, but I'm facing difficulty. Then, here is my question.

Question : Could you show me how to prove that $\sum_{j=1}^{n}\frac{1}{a+jd}$ is not an integer for any integers $a\gt1, d\gt0$.