Found this question in a worksheet for induction problems, and for the life of me I can't figure out how to prove this via induction. First, I tried playing around with it, proving some more specific examples, such as if $a_1,a_2$ $\in \mathbb{Z}^+$, then
$(a_1+a_2)(\frac{1}{a_1}+\frac{1}{a_2})\ge 4$,
with success. However, I'm unsure how to proceed with the much more generalized question, namely how $a_1,a_2,a_3,\cdots, a_n$ $\in \mathbb{R}$ and an $n$ number of additions change the problem. Here is my shoddy attempt at a start of an induction proof on the statement:
Let $S(n): (\sum_{i=1}^{n}a_i )(\sum_{i=1}^{n}\frac{1}{a_i})\ge n^2$. We proceed with induction first by considering $S(1)$.
$S(1):$ $a_1 \cdot \frac{1}{a_1} = 1 \ge 1^2$, which obviously holds true.
Then we assume for some $k< n\in \mathbb{Z}$, that $S(k)$ is true.
$S(k): (\sum_{i=1}^{k}a_i )(\sum_{i=1}^{k}\frac{1}{a_i})\ge k^2$.
Then we have to show that $S(k)\implies S(k+1) : (\sum_{i=1}^{k+1}a_i )(\sum_{i=1}^{k+1}\frac{1}{a_i})\ge (k+1)^2$.
This is where I get stuck. I have no clue how to continue at all, this isn't similar to any type of induction question I've ever come across, combining sums and inequalities. I'm assuming that there is definitely a way to manipulate the sums themselves so that we can invoke the induction hypothesis, however I'm either too tired or stupid to see it. Any help will be appreciated.
Best Answer
For an inductive proof, write $$ \sum_1^{n+1}a_i = a_{n+1} +\sum_1^n a_i$$ and $$\sum_1^{n+1}\frac1{a_i}=\frac1{a_{n+1}}+\sum_1^n\frac1{a_i}.$$ Multiply out, obtaining four terms. One of the four terms exceeds $n^2$, by the induction hypothesis. One of them equals $1$. What about the remaining two terms? If you can show the sum of these two terms exceeds $2n$, you are done.
Indeed, the remaining terms are $$a_{n+1}\sum_1^n\frac1{a_i} + \frac1{a_{n+1}}\sum_1^n a_i $$ which can be packaged together as the sum of $n$ terms of the form $u_i+\frac 1{u_i}$. Assuming each of the $a$'s is positive, this means each $u_i$ is positive. Your job is now to show that $u_i+\frac1{u_i}\ge 2$ for every $i$.