Sum of reciprocal of the terms of an arithmetic progression:

Let $a$ and $d$ be two coprime integers. (The great common divisor between $a$ and $d$ is $1$ ). Prove that: $(\forall n\in\mathbb{N}^*):\frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2d}+\cdots+\frac{1}{a+nd}\notin\mathbb{Z}$ .

#Proofs #MathProblem #Math

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Why is $\gcd(a,d)=1$ ? If this holds, then it also holds for $a,d$ with $\gcd(a,d)>1$ .

Tim Vermeulen - 7 years, 9 months ago

You could edit title as Sum of an Harmonic Progression....

Rahul Nahata - 7 years, 9 months ago

What about $a=1$ , $d=1$ , and $n=\infty$ ? Or is that not allowed? EDIT: would $\infty\in\mathbb{Z}$ ?

Daniel Liu - 7 years, 9 months ago

$\infty \not\in \mathbb{N} \implies \infty \not\in \mathbb{Z}.$

Tim Vermeulen - 7 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$