# How to show that $\frac{R_1R_2}{R_1+r_2}<(R_1,R_2)$ strictly using AM-GM inequality?

I was reading about parallel circuits in Physics.Equivalent resistance of $n$ resistors in parallel is given by $\displaystyle\frac1{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+...+\frac{1}{R_n}$.

I tried to prove that $R_{eq}$ will always be less than $R_1,R_2,...,R_n$.

I tried to prove it for two resistors,where $R_{eq}=\frac{R_1R_2}{R_1+R_2}$.

By applying AM-GM on $R_1,R_2$we have,

$\frac{R_1R_2}{4}\geq \frac{R_1R_2}{R_1+R_2}$.

Now,I have no idea how to show from here that $(R_1,R_2)\geq\frac{R_1R_2}{R_1+R_2}$ and how it can be extended for $R_n$

Thanks for any help!!

• FYI: The harmonic mean of a set of numbers $\{a_n\}$ is $H=\left(\sum_n \frac{1}{a_n}\right)^{-1}$. – Semiclassical Jun 3 '16 at 17:12
• $=n\sum_n\Bigl(\frac1{a_n}\Bigr)^{-1}$ more exactly. – Bernard Jun 3 '16 at 17:15
• This is not mathematical physics. Edited tags. – user_of_math Jun 3 '16 at 17:27
• $\displaystyle{{1 \over R_{\mathrm{eq}}} > {1 \over R_{k}}\,,\quad\forall\ k}$. – Felix Marin Jun 4 '16 at 5:22

That is more trivial. If $$\frac{1}{R_{eq}}=\sum_{k=1}^{n}\frac{1}{R_k}$$ obviously $$\frac{1}{R_{eq}}> \frac{1}{R_k}$$ for any $k\in\{1,2,\ldots,n\}$, hence $R_{eq}< R_k$, so $$R_{eq} < \min_{k} R_k.$$
• @tatan: if $u = a+b+c+\ldots$ and $a,b,c,\ldots >0$, then $u>a, u>b, u>c,\ldots$. If that isn't trivial... – Jack D'Aurizio Jun 3 '16 at 17:39
suppose $1/R_1 = x$ and $1/R_2=y$ and $1/R_{eq}=z$ now we know that $z=x+y$ and $x>0,y>0$
therefore $z>x$ and $z>y$ and hence $1/R_{eq}>1/R_1$ and $1/R_{eq}>1/R_2$
and therefore $R_{eq}<R_1$ and $R_{eq}<R_2$
Suppose we remove one of the resistors $R_k$ from the circuit. Then the new equivalent resistance will $\left(\frac{1}{R_{eq}}-\frac{1}{R_k}\right)^{-1}>R_{eq}$ i.e. the equivalent resistance strictly increases. If we repeat this until only one resistor $R_f$ remains, then $R_f$ is the final equivalent resistance. Hence each individual resistor must have a larger resistance than the equivalent resistance of them in parallel.